Macro jfv

Macroeconomics: an Introduction
Jes´us Fern´andez-Villaverde
University of Pennsylvania
1

The Scope of Macroeconomics
• Microeconomics: Object of interest is a single (or small number of)
household or ﬁrm.
• Macroeconomics: Object of interest is the entire economy. We care
mostly about:
1. Growth.
2. Fluctuations.
2

Relation between Macro and Micro
• Micro and Macro are consistent applications of standard neoclassical
theory.
• Unifying theme, EQUILIBRIUM APPROACH:
1. Agents optimize given preferences and technology.
2. Agents’ actions are compatible with each other.
• This requires:
1. Explicit about assumptions.
2. Models as abstractions.
3

What are the Requirements of Theory?
• Well articulated models with sharp predictions.
• Good theory cannot be vague: predictions must be falsiﬁable.
• Internal Consistency.
• Models as measurement tools.
All this is Scientiﬁc Discipline.
4

Why should we care about Macroeconomics?
• Self Interest: macroeconomic aggregates aﬀect our daily life.
• Cultural Literacy: understanding our world.
• Common Welfare: Essential for policymakers to do good policy.
• Civic Responsibility: Essential for us to understand our politicians.
5

A Brief Overview of the History of Macroeconomics I
• Classics (Smith, Ricardo, Marx) did not have a sharp distinction be-
tween micro and macro.
• Beginning of the XX century: Wicksell, Pigou.
• J.M. Keynes, The General Theory of Employment, Interest, and Money
(1936).
• 1945-1970, heyday of Neoclassical Synthesis: Samuelson, Solow, Klein.
• Monetary versus Fiscal Policy: Friedman, Tobin.
6

A Brief Overview of the History of Macroeconomics II
• 1972, Rational Expectations Revolution: Lucas, Prescott, Sargent.
• 1982, Real Business Cycles: Kydland and Prescott.
• 1990’s, Rich dynamic equilibrium models.
• Future?
7

Why do Macroeconomist Disagree?
• Most research macroeconomist agree on a wide set of issues.
• There is wide agreement on growth theory.
• There is less agreement on business cycle theory.
• Normative issues.
• Are economist ideologically biased? Caplan (2002).
8

National Income and Product Accounts
1

A Guide to NIPA’s
• What is the goal?
• When did it begin? Role of Simon Kuznets:
1. Nobel Prize in Economics 1971.
2. Prof. at Penn during the key years of NIPA creation.
• Gigantic intellectual achievement.
• Elaborated by Bureau of Economic Analysis and published in the Sur-
vey of Current Business. http://www.bea.gov/
2

Question: How are macroeconomic aggregates measured?
3

Gross Domestic Product (GDP)
Can be measured in three diﬀerent, but equivalent ways:
1. Production Approach.
2. Expenditure Approach.
3. Income Approach.
4

Nominal GDP
• For 2003, nominal GDP was:
$11, 004, 000, 000, 000
• Population, July 2003 was:
290, 788, 976
• Nominal GDP per capita is roughly:
$37, 842
5

Computing GDP through Production
• Calculate nominal GDP by adding value of production of all industries:
production surveys.
• Problem of double-counting: i.e. USX and GM.
• Value Added=Revenue−Intermediate Goods.
• Nominal GDP=Sum of Value Added of all Industries.
6

Computing GDP through Expenditure
C = Consumption
I = (Gross Private) Investment
G = Government Purchases
X = Exports
M = Imports
Y = Nominal GDP
Y ≡ C + I + G + (X − M)
7

Consumption (C)
• Durable Goods: 3 years rule.
• Nondurable Goods.
• Services.
8

Gross Private Investment (I)
• Nonresidential Fixed Investment.
• Residential Fixed Investment.
• Inventory Investment.
• Stocks vs. Flows.
9

Investment and the Capital Stock
• Capital Stock: total amount of physical capital in the economy
• Depreciation: the part of the capital stock that wears out during the
period
• Capital Stock at end of this period=Capital Stock at end of last
period+Gross Investment in this period−Depreciation in this period
• Net Investment=Gross Investment−Depreciation=Capital Stock, end
this per.−Capital Stock, end of last per.
10

Inventory Investment
• Why included in GDP?
• Inventory Investment=Stock of Inventories at end of this year−Stock
of Inventories at the end of last year
• Final Sales=Nominal GDP−Inventory Investment
11

Government Purchases (G)
• Sum of federal, state, and local purchases of goods and services.
• Certain government outlays do not belong to government spending:
transfers (SS and Interest Payments).
• Government Investment.
12

Exports (E) and Imports (M)
• Exports: deliveries of US goods and services to other countries.
• Imports: deliveries of goods and services from other countries to the
US.
• Trade Balance=Exports−Imports
• Trade Deﬁcit: if trade balance negative.
• Trade Surplus: if trade balance positive
13

Composition of GDP - Spending in billion $ in % of GDP
Total Nom. GDP 11,004.0 100.0%
Consumption 7,760.0 70.5%
Durable Goods
Nondurable Goods
Services
950.7
2,200.1
4,610.1
8.6%
20.0%
41.9%
Gross Private Investment 1,667.0 15.1%
Nonresidential
Residential
Changes in Inventory
1,094.7
572.3
−1.2
9.9%
5.2%
−0.0%
Government Purchases 2,075.5 18.9%
Federal Gov.
State & Local Gov.
752,2
1,323.3
6.8%
12.2%
Net Exports -498.1 -4.5%
Exports
Imports
1,046.2
1,544.3
9.5%
14.0%
Gross National Product 11,059.2 100.5%
14

Computing GDP through Income
National Income: broadest measure of the total incomes of all Americans
Gross Domestic Product (11,004.0)
+Factor Inc. from abroad (329.0) − Factor Inc. to abroad (273.9)
= Gross National Product (11,059.2)
−Depreciation (1,359.9)
= Net National Product (9,705.2)
−Statistical Discrepancy (25.6)
= National Income (9,679.6)
15

Distribution of National Income
1. Employees’ Compensation: wages, salaries, and fringe beneﬁts.
2. Proprietors’ Income: income of noncorporate business.
3. Rental Income: income that landlords receive from renting, including
“imputed” rent less expenses on the house, such as depreciation.
4. Corporate Proﬁts: income of corporations after payments to their
workers and creditors.
5. Net interest: interests paid by domestic businesses plus interest earned
from foreigners.
16

Labor and Capital Share
• Labor share: the fraction of national income that goes to labor income
• Capital share: the fraction of national income that goes to capital
income.
• Labor Share= Labor Income
National Income
• Capital Share= Capital Income
National Income
• Proprietor’s Income?
17

Distribution of National Income
Billion $US % of Nat. Inc.
National Income less Prod. Tax. 8,841.0 100.0%
Compensation of Employees 6,289.0 71.1%
Proprietors’ Income 834.1 9.4%
Rental Income 153.8 1.7%
Corporate Proﬁts 1021.1 11.6%
Net Interest 543.0 6.1%
18

Composition of National Income
Industries Val. Add. in %
National Income’ 9,396.6 100.0%
Agr., Forestry, Fish. 75.8 0.8%
Mining 94.9 1.0%
Construction 476.5 5.1%
Manufacturing 1,113.1 11.8%
Public Utilities 156.0 1.7%
Transportation 259.9 2.8%
Wholesale Trade 569,6 6.1%
Retail Trade 752.8 7.7%
Fin., Insur., Real Est. 1,740.8 18.5%
Services 1,893.6 20.1%
Government 1,182.8 12.6%
Rest of the World 55.1 0.6%
19

Other Income Concepts: Personal Income
• Income that households and noncorporate businesses receive
National Income (9,679.6) − Corporate Proﬁts (1021.1)
−Net Taxes on Production and Imports (751.3)
−Net Interest (543.0) − Contributions for Social Insurance (773.1)
+Personal Interest Income (1,322.7)
+Personal Current Transfer Receipts (1,335.4)
= Personal Income (9,161.8)
20

Other Income Concepts: Disposable Personal Income
• Income that households and noncorporate businesses can spend, after
having satisﬁed their tax obligations
Personal Income (9,161.8)
−Personal Tax and Nontax Payments (1,001.9)
= Disposable Personal Income (8,159.9)
21

Investment and Saving
• Private Saving (S): gross income minus consumption and taxes plus
transfers
• From income side Y = C + S + T − TR + NFP
• From expenditure side Y = C + I + G + X − M
I|{z} =
Private Inv.
S|{z}
Private Saving
+ T − TR − G| {z }
Public Saving
+ M − X + NFP| {z }
Foreign Saving
22

Some Nontrivial Issues
• Releases of Information and revisions.
• Methodological Changes.
• Technological Innovation.
• Underground Economy.
• Non-market activities.
• Welfare.
23

Price Indices
Question: How to compute the price level?
Idea: Measure price of a particular basket of goods today versus price of
same basket in some base period
24

Example: Economy with 2 goods, hamburgers and coke
ht = # of hamburgers produced, period t
pht = price of hamburgers in period t
ct = # of coke produced, period t
pct = price of coke in period t
(h0, ph0, c0, pc0) = same variables in period 0
Laspeyres price index
Lt =
phth0 + pctc0
ph0h0 + pc0c0
Paasche price index
Pat =
phtht + pctct
ph0ht + pc0ct
25

Problems with Price Indices
• Laspeyres index tends to overstate inﬂation.
• Paasche index tends to understate inﬂation.
• Fisher Ideal Index: geometric mean: (Lt × Pat)0.5.
• Chain Index.
26

From Nominal to Real GDP
• Nominal GDP: total value of goods and services produced.
• Real GDP: total production of goods and services in physical units.
• How is real GDP computed in practice, say in 2004?
27

1. Pick a base period, say 1996
2. Measure dollar amount spent on hamburgers.
3. Divide by price of hamburgers in 2004 and multiply by price in 1996.
(this equals the number of hamburgers sold in 2004, multiplied by the
price of hamburgers in 1996 -the base period).
4. Sum over all goods and services to get real GDP.
28

For our example ...
Nominal GDP in 2004 = h2004ph2004 + c2004pc2004
Real GDP in 1996 = h2000ph1996 + c2004pc1996
Note that
GDP deﬂator =
Nominal GDP
Real GDP
=
h2004ph2004 + c2004pc2004
h2004ph1996 + c2004pc1996
29

Measuring Inﬂation I
• πt =
Pt−Pt−1
Pt−1
where Pt is the “Price Level”.
• GDP deﬂator: basket corresponds to current composition of GDP.
• Consumer Price Index (CPI): basket corresponds to what a typical
household bought during a typical month in the base year
CPI =
h1996ph2004 + c1996pc2004
h1996ph1996 + c1996pc1996
• CPI important because of COLA’s.
30

Measuring Inﬂation II
• CPI may overstate inﬂation: Boskin Commission, New Goods.
• How to measure new technologies? David Cutler’s example:
1. Average heart attack in mid-1980’s costs about $199912,000 to
treat.
2. Average heart attack in late 1990’s costs about $199922,000 to
treat.
3. Average life expectancy in late 1990’s is one year higher than in
mid-1980’s.
4. Is health care more expensive now?
31

Inﬂation over History I
• How much is worth $1 from 1789 in 2003?
1. $20.76 using the Consumer Price Index
2. $21.21 using the GDP deﬂator.
32

Inﬂation over History II
33

More on Growth Rates
• Growth rate of a variable Y (say nominal GDP) from period t − 1 to
t is given by
gY (t − 1, t) =
Yt − Yt−1
Yt−1
• Growth rate between period t − 5 and period t is given by
gY (t − 5, t) =
Yt − Yt−5
Yt−5
34

• Suppose that GDP equals Yt−1 in period t − 1 and it grows at rate
gY (t − 1, t). How big is GDP in period t?
gY (t − 1, t) =
Yt − Yt−1
Yt−1
gY (t − 1, t) ∗ Yt−1 = Yt − Yt−1
gY (t − 1, t) ∗ Yt−1 + Yt−1 = Yt
(1 + gY (t − 1, t))Yt−1 = Yt
Hence GDP in period t equals GDP in period t − 1, multiplied by 1
plus the growth rate.
• Example: If GDP is $1000 in 2004 and grows at 3.5%, then GDP in
2005 is
Y2005 = (1 + 0.035) ∗ $1000 = $1035
35

• Suppose GDP grows at a constant rate g over time. Suppose at period
0 GDP equals some number Y0 and GDP grows at a constant rate of
g% a year. Then in period t GDP equals
Yt = (1 + g)tY0
• Example: If Octavio Augustus would have put 1 dollar in the bank at
year 0AD and the bank would have paid a constant real interest rate
of 1.5%, then in 2000 he would have:
Y2000 = (1.015)2000 ∗ $1 = $8, 552, 330, 953, 000
which is almost the US GDP for last year.
36

• Reverse question: Suppose we know GDP at 0 and at t. Want to know
at what constant rate GDP must have grown to reach Yt, starting from
Y0 in t years.
Yt = (1 + g)tY0
(1 + g)t =
Yt
Y0
(1 + g) =
Ã
Yt
Y0
!1
t
g =
Ã
Yt
Y0
!1
t
− 1
37

• Example: In 1900 a country had GDP of $1,000 and in 2000 it has
GDP of $15,000. Suppose that GDP has grown at constant rate g.
How big must this growth rate be? Take 1900 as period 0, 2000 as
period 100, then
=
Ã
$15, 000
$1, 000
! 1
100
− 1
= 0.027 = 2.7%
38

• Question: We know GDP of a country in period 0 and its growth rate
g. How many time periods it takes for GDP in this country to double
(to triple and so forth).
Yt = (1 + g)tY0
(1 + g)t =
Yt
Y0
Since log
³
ab
´
= b ∗ log(a)
log
³
(1 + g)t
´
= log
Ã
Yt
Y0
!
t ∗ log(1 + g) = log
Ã
Yt
Y0
!
t =
log
³
Yt
Y0
´
log(1 + g)
39

• Suppose we want to ﬁnd the number of years it takes for GDP to
double, i.e. the t such Yt
Y0
= 2. We get
t =
log(2)
log(1 + g)
• Example: with g = 1% it takes 70 years, with g = 2% it takes 35
years, with g = 5% it takes 14 years.
40

Transactions with the Rest of the World
Trade Balance = Exports − Imports
Current Account Balance = Trade Balance + Net Unilateral Transfers
• Unilateral transfers: include aid to poor countries, interest payments
to foreigners for US government debt, and grants to foreign researchers
or institutions.
41

• Net wealth position of the US: diﬀerence between what the US is owed
and what it owes to foreign countries.
• Capital account balance: equals to the change of the net wealth posi-
tion of the US
Capital Account Balance this year
= Net wealth position at end of this year
−Net wealth position at end of last year
42

Unemployment Rate
• Labor force: number of people, 16 or older, that are either employed
or unemployed but actively looking for a job.
• Current Population Survey.
• Unemployment Rate= number of unemployed people
labor force
• What is the current unemployment rate now?
43

Interest Rates
• Important as they determine how costly it is to borrow money
• Suppose in period t − 1 you borrow the amount $Bt−1. The loan
speciﬁes that in period t you have to repay $Bt. Nominal interest rate
on the loan from period t − 1 to period t, it, is
it =
Bt − Bt−1
Bt−1
• Real interest rate rt
rt = it − πt
44

• Example: In 2003 you borrow $15, 000 and the bank asks you to repay
$16, 500 exactly one year later. The yearly nominal interest rate from
2003 to 2004 is
i2004 =
$16, 500 − $15, 000
$15, 000
= 0.1 = 10%
Now suppose the inﬂation rate is 3% in 2004. Then the real interest
rate equals 10% − 3% = 7%.
45

Introduction to Growth Theory
1

Growth Theory
I do not see how one can look at ﬁgures like these without seeing them
as representing possibilities. Is there some action a government could take
that would lead the Indian economy to grow like Indonesia’s or Egypt’s?
If so, what exactly? If not, what is it about the “nature of India” that
makes it so? The consequences for human welfare involved in questions
like these are simply staggering: Once one starts to think about them, it
is hard to think about anything else (Lucas 1988, p. 5).
2

Some Motivation
• Diﬀerences across countries:
1. Out of 6.4 billion people, 0.8 do not have access to enough food,
1 to safe drinking water, and 2.4 to sanitation.
2. Life expectancy in rich countries is 77 years, 67 years in middle
income countries, and 53 million in poor countries.
• Diﬀerences across time:
1. Japanese boy born in 1880 had a life expectancy of 35 years, today
81 years.
2. An American worked 61 hours per week in 1870, today 34.
3

History of Economic Growth Theory: a Roadmap
1. Smith, Ricardo, Malthus and Mill had little hope for sustained growth.
2. Forgotten for a long while. Ill attempted in UK (Harrod and Domar).
3. Robert Solow (MIT, Nobel 1987): two main papers: 1956 and 1957.
4. Completed by David Cass (Penn) and Tjalling Koopmans (Nobel 1971).
5. 80’s and 90’s: Paul Romer (Stanford, Nobel 20??) and Robert Lucas
(Chicago, Nobel 1995).
4

Growth Facts (Nicholas Kaldor)
Stylized growth facts (empirical regularities of the growth process) for the
US and for most other industrialized countries
1. Output (real GDP) per worker y = Y
L and capital per worker k = K
L
grow over time at relatively constant and positive rate.
2. They grow at similar rates, so that the ratio between capital and
output, K
Y is relatively constant over time
3. The real return to capital r (and the real interest rate r−δ) is relatively
constant over time.
4. The capital and labor shares are roughly constant over time.
5

U.S. Real GDP per Worker (1995
Prices), 1890-1995
$0
$10
$20
$30
$40
$50
$60
$70
1890 1920 1950 1980
Thousands per Worker

Data
• How do incomes and growth rates vary across countries.
• Summers-Heston data set at Penn: follow 104 countries over 30 years.
• Focus on income (GDP) per worker.
• Measure income (GDP) using PPP-based exchange rates.
6

Development Facts I
1. Enormous variation of per worker income across countries.
2. Enormous variation in growth rates of per worker income across coun-
tries.
Growth “Miracles” g60−97
South Korea 5.9%
Taiwan 5.2%
Growth “Disasters”
Venezuela -0.1%
Madagascar -1.4%
7

0 0.2 0.4 0.6 0.8 1 1.2 1.4
0
5
10
15
20
25
30
35
40
Distribution of Relative Per Worker Income
Income Per Worker Relative to US
NumberofCountries
1960
1990

1997 PPP GDP per Capita
$0
$10,000
$20,000
$30,000
Country
1997 PPP GDP per Capit
$0
$10,000
$20,000
$30,000
Country

Output per Capita as a Share of US
Level
0%
25%
50%
75%
100%
1950 1960 1970 1980 1990 2000
Years
Canada
USA
Japan
France
Germany
(W)
Italy
Britain

−0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05 0.06
0
5
10
15
20
25
Distribution of Average Growth Rates (Real GDP) Between 1960 and 1990
Average Growth Rate
NumberofCountries

Development Facts II
3. Growth rates are not constant over time for a given country.
4. Countries change their relative position in the international income
distribution.
8

Development Facts III
5. Growth in output and growth in international trade are closely related.
6. Demograhic transition.
7. International migration.
8. “La longue dur´ee”.
9

Year Population* GDP per Capita**
-5000 5 $130
-1000 50 $160
1 170 $135
1000 265 $165
1500 425 $175
1800 900 $250
1900 1625 $850
1950 2515 $2030
1975 4080 $4640
2000 6120 $8175
*Millions
**In year-2000 international dollars.

Population Growth Since 1000
World Population Since 1000
0
1000
2000
3000
4000
5000
6000
7000
1000 1200 1400 1600 1800 2000
Year

Stylized Picture of the Demographic Transition
The Demographic Transition
Time
Birth Rate
Death
Rate
Rate of
Natural
Increase
Onset of the
demographic
transition
Moment of
maximum
increase
End of the
transition

Growth Accounting
1

Growth Accounting
• Output is produced by inputs capital K and labor L, in combination
with the available technology A
• Want: decompose the growth rate of output into the growth rate of
capital input, the growth rate of labor input and technological progress.
This exercise is called growth accounting.
• Why?
2

Aggregate production function
• Maps inputs into output:
Y = F (A, K, L)
A is called total factor productivity (TFP).
• Cobb-Douglas example:
Y = AKαL1−α
• Interpretation.
3

Discrete vs. Continuous Time
• In discrete time a variable is indexed by time: xt.
• In continuous time a variable is a function of time: x(t).
• We observe the world only in discrete time...
• but it is often much easier to work with continuous time!
4

Growth Rates and Logarithms I
• Remember:
gx(t − 1, t) =
xt − xt−1
xt−1
1 + gx(t − 1, t) =
xt
xt−1
• Take logs on both sides:
log (1 + gx(t − 1, t)) = log
Ã
xt
xt−1
!
5

Growth Rates and Logarithms II
• Taylor series expansion of log (1 + y) around y = 0:
log (1 + y)|y=0 = ln 1 +
1
1!
y + higher order terms ' y
• Then:
log (1 + gx(t − 1, t)) ' gx(t − 1, t) ' log
Ã
xt
xt−1
!
gx(t − 1, t) ' log xt − log xt−1 = ∆ log xt
• Remember from calculus that validity of Taylor series expansion is
local: g small!
6

Moving between Continuous and Discrete Time I
• Let x(t) be a variable that depends of t.
• Notation:
·
x(t) ≡
dx(t)
dt
• Take log x(t). Then:
d log((x(t))
dt
=
·
x(t)
x(t)
= gx(t)
• Why is this useful?
7

Moving between Continuous and Discrete Time II
• The deﬁnition of time derivative is:
·
x(t) = lim
∆t→0
x(t + ∆t) − x(t)
∆t
• Then:
·
x(t)
x(t)
=
lim∆t→0
x(t+∆t)−x(t)
∆t
x(t)
• When ∆t is small (let’s say a year):
gx(t) =
·
x(t)
x(t)
'
x(t + 1) − x(t)
x(t)
= gx(t − 1, t) ' ∆ log xt
8

Growth Rates of Ratios I
• Suppose k(t) =
K(t)
L(t)
. What is gk(t)?
• Step 1: take logs
log(k(t)) = log(K(t)) − log(L(t))
• Step 2: diﬀerentiate with respect to time
d log((k(t))
dt
=
d log(K(t))
dt
−
d log(L(t))
dt
˙k(t)
k(t)
=
˙K(t)
K(t)
−
˙L(t)
L(t)
gk(t) = gK(t) − gL(t)
9

Growth Rates of Ratios II
• Growth rate of a ratio equals the diﬀerence of the growth rates:
gk(t) = gK(t) − gL(t)
• Ratio constant over time requires that both variables grow at same
rate:
gk(t) = 0 ⇒ gK(t) = gL(t)
10

Growth Rates of Weighted Products I
• Suppose
Y (t) = K(t)αL(t)1−α
What is gY (t)?
• Step 1: take logs
log(Y (t)) = α log(K(t)) + (1 − α) log(L(t))
11

Growth Rates of Weighted Products II
• Step 2: diﬀerentiate
d log(Y (t))
dt
= α
d log(K(t))
dt
+ (1 − α)
d log(L(t))
dt
˙Y (t)
Y (t)
= α
˙K(t)
K(t)
+ (1 − α)
˙L(t)
L(t)
gY (t) = αgK(t) + (1 − α)gL(t)
• Growth rate equals weighted sum, with weights equal to the share
parameters
12

Growth Accounting I
• Observations in discrete time.
• Production Function: Y (t) = F (A(t), K(t), L(t))
• Diﬀerentiating with respect to time and dividing by Y (t)
˙Y (t)
Y (t)
=
FAA(t)
Y (t)
˙A(t)
A(t)
+
FkK(t)
K(t)
˙K(t)
K(t)
+
FLL(t)
Y (t)
˙L(t)
L(t)
13

Growth Accounting II
• Useful benchmark: Cobb-Douglas Y (t) = A(t)K(t)αL(t)1−α.
• Why?
• Taking logs and diﬀerentiating with respect to time gives
gY (t) = gA(t) + αgK(t) + (1 − α)gL(t)
• gA is called TFP growth or multifactor productivity growth.
14

Doing the Accounting
• Pick an α (we will learn that α turns out to be the capital share).
• Measure gY , gK and gL from the data.
• Compute gA as the residual:
gA(t) = gY (t) − αgK(t) − (1 − α)gL(t)
• Therefore gA is also called the Solow residual.
• Severe problems if mismeasurement (gK is hard to measure).
15

Data for the US
• We pick α = 1
3
Per. gY αgK (1 − α)gL TFP (gA)
48 − 98 2.5 0.8 (32%) 0.2 (8%) 1.4 (56%)
48 − 73 3.3 1.0 (30%) 0.2 (6%) 2.1 (64%)
73 − 95 1.5 0.7 (47%) 0.3 (20%) 0.6 (33%)
95 − 98 2.5 0.8 (32%) 0.3 (12%) 1.4 (56%)
• Key observation: Productivity Slowdown in the 70’s
• Note: the late 90’s look much better
16

Reasons for the Productivity Slowdown
1. Sharp increases in the price of oil in 70’s
2. Structural changes: more services and less and less manufacturing
goods produced
3. Slowdown in resources spent on R&D in the late 60’s.
4. TFP was abnormally high in the 50’s and 60’s
5. Information technology (IT) revolution in the 70’s
17

Growth Accounting for Other Countries
• One key question: was fast growth in East Asian growth miracles
mostly due to technological progress or mostly due to capital accumu-
lation?
• Why is this an important question?
18

Country Per. gY α αgK (1 − α)gL gA
Germany 60-90 3.2 0.4 59% −8% 49%
Italy 60-90 4.1 0.38 49% 3% 48%
UK 60-90 2.5 0.39 52% −4% 52%
Argentina 40-80 3.6 0.54 43% 26% 31%
Brazil 40-80 6.4 0.45 51% 20% 29%
Chile 40-80 3.8 0.52 34% 26% 40%
Mexico 40-80 6.3 0.63 41% 23% 36%
Japan 60-90 6.8 0.42 57% 14% 29%
Hong Kong 66-90 7.3 0.37 42% 28% 30%
Singapore 66-90 8.5 0.53 73% 31% −4%
South Korea 66-90 10.3 0.32 46% 42% 12%
Taiwan 66-90 9.1 0.29 40% 40% 20%
19

Neoclassical Growth Model
1

Models and Assumptions I
• What is a model? A mathematical description of the economy.
• Why do we need a model? The world is too complex to describe it in
every detail. A model abstracts from details to understand clearly the
main forces driving the economy.
• What makes a model successful? When it is simple but eﬀective in
describing and predicting how the world works.
2

Models and Assumptions II
• A model relies on simplifying assumptions. These assumptions drive
the conclusions of the model. When analyzing a model it is crucial to
spell out the assumptions underlying the model.
• Realism may not a the property of a good assumption.
• An assumption is good when it helps us to build a model that accounts
for the observations and predicts well.
3

Basic Assumptions of the Neoclassical Growth Model
1. Continuous time.
2. Single good in the economy produced with a constant technology.
3. No government or international trade.
4. All factors of production are fully employed.
5. Labor force grows at constant rate n =
˙L
L.
6. Initial values for capital, K0 and labor, L0 given.
4

Production Function I
• Neoclassical (Cobb-Douglas) aggregate production function:
Y (t) = F(K(t), L(t)) = K(t)αL(t)1−α
• To save on notation write:
Y = KαL1−α
where the dependency on t is understood implicitly.
5

Properties of the Technology I
• Constant returns to scale:
λY = (λK)α
(λL)1−α
= λKαL1−α
• Inputs are essential:
F (0, 0) = F (K, 0) = F (0, L) = 0
• Marginal productivities are positive:
∂F
∂K
= αAKα−1L1−α > 0
∂F
∂L
= (1 − α) AKαL−α > 0
6

Properties of the Technology II
• Marginal productivities are decreasing,
∂2F
∂K2
= (α − 1) αKα−2L1−α < 0
∂2F
∂L2
= (α − 1) αKαL−α−1 < 0
• Inada Conditions,
lim
K→0
αKα−1L1−α = ∞, lim
K→∞
αKα−1L1−α = 0
lim
L→0
(1 − α) KαL−α = ∞, lim
L→∞
(1 − α) KαL−α = 0
7

Per Worker Terms
• Deﬁne x = X
L as a per worker variable.
• Then
y =
Y
L
=
KαL1−α
L
=
µ
K
L
¶a µ
L
L
¶1−α
= kα
• Per worker production function has decreasing returns to scale.
8

Capital Accumulation I
• Capital accumulation equation:
˙K = sY − δK
• Important additional assumptions:
1. Constant saving rate
2. Constant depreciation rate
9

Capital Accumulation II
• Dividing by K in the capital accumulation equation:
˙K
K
= s
Y
K
− δ
• Some Algebra:
˙K
K
= s
Y
K
− δ = s
Y
L
K
L
− δ = s
y
k
− δ
10

Capital Accumulation III
• Now remember that:
˙k
k
=
˙K
K
−
˙L
L
=
˙K
K
− n ⇒
˙K
K
=
˙k
k
+ n
• We get
˙k
k
+ n = s
y
k
− δ ⇒ ˙k = sy − (δ + n) k
• Fundamental Diﬀerential Equation of Neoclassical Growth Model:
˙k = skα − (δ + n) k
11

Graphical Analysis
• Change in k, ˙k is given by diﬀerence of skα and (δ + n)k
• If skα > (δ + n)k, then k increases.
• If skα < (δ + n)k, then k decreases.
• Steady state: a capital stock k∗ where, when reached, ˙k = 0
• Unique positive steady state in Neoclassical Growth model.
• Positive steady state (locally) stable.
12

Close-Form Solution I
• ˙k = skα − (δ + n) k is a Bernoulli Equation.
• Change of variable:
z = k1−α
• Then:
˙z = (1 − α) k−α ˙k ⇒ ˙k = ˙z (1 − α)−1 kα
13

Close-Form Solution II
• Some algebra
˙z (1 − α)−1
kα = skα − (δ + n) k
˙z (1 − α)−1
= s − (δ + n) k1−α = s − (δ + n) z
˙z = (1 − α) s − (1 − α) (δ + n) k1−α
˙z + λz = (1 − α) s
where λ = (1 − α) (δ + n).
14

Close-Form Solution III
• We have a linear, first order differential equation with constant coef-
ficients.
• Integrating with respect to eλtdt:
Z
( ˙z + λz) eλtdt =
Z
(1 − α) seλtdt
we get
zeλt =
(1 − α) s
λ
eλt + b
where b is an integrating constant.
15

Close-Form Solution IV
• Then:
z(t) =
(1 − α) s
λ
+ be−λt
• Substituting back: z = k1−α we get the general solution:
k(t) =
µ
s
δ + n
+ be−λt
¶ 1
1−α
• To ﬁnd the particular solution note that
z(0) =
(1 − α) s
λ
+ be−λ0 =
s
δ + n
+ b = z0 ⇒ b = z0 −
s
δ + n
16

Close-Form Solution V
• Then:
z(t) =
s
δ + n
+
µ
z0 −
s
δ + n
¶
e−λt
• Interpretation of λ.
• Substituting back z = k1−α we get:
k(t) =
³
s
δ+n +
³
k1−α
0 − s
δ+n
´
e−λt
´ 1
1−α
y(t) =
³
s
δ+n +
³
k1−α
0 − s
δ+n
´
e−λt
´ α
1−α
17

Steady State Analysis
• Steady State: ˙k = 0
• Solve for steady state
0 = s (k∗)α
− (n + δ)k∗ ⇒ k∗ =
µ
s
n + δ
¶ 1
1−α
• Steady state output per worker y∗ =
³
s
n+δ
´ α
1−α
• Steady state output per worker depends positively on the saving (in-
vestment) rate and negatively on the population growth rate and de-
preciation rate.
18

Comparative Statics
• Suppose that of all a sudden saving rate s increases to s0 > s. Suppose
that at period 0 the economy was at its old steady state with saving
rate s.
• (n + δ)k curve does not change.
• skα = sy shifts up to s0y.
• New steady state has higher capital per worker and output per worker.
• Monotonic transition path from old to new steady state.
19

Evaluating the Basic Neoclassical Growth Model: the Good
• Why are some countries rich (have high per worker GDP) and others
are poor (have low per worker GDP)?
• Neoclassical Growth model: if all countries are in their steady states,
then:
1. Rich countries have higher saving (investment) rates than poor
countries.
2. Rich countries have lower population growth rates than poor coun-
tries.
• Data seem to support this prediction of the Neoclassical Growth model.
20

GDP per Worker 1990 as Function of Investment Rate
Average Investment Share of Output 1980−90
GDPperWorker1990in$10,000
−0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5

Growth Rates and Investment Rates
Average Investment Rate 1980−1990
AverageGrowthRate1960−1990
−0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
−0.04
−0.02
0
0.02
0.04
0.06

GDP per Worker 1990 as Function of Population Growth Rate
Average Population Growth Rate 1980−90
GDPperWorker1990in$10,000
0 0.01 0.02 0.03 0.04 0.05
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5

Evaluating the Basic Neoclassical Growth Model: the Bad
• Are saving and population growth rates exogenous?
• Are the magnitude of diﬀerences created by the model right?
yus
ss =
µ
0.2
0.01 + 0.06
¶1
2
= 1.69
ychad
ss =
µ
0.05
0.02 + 0.06
¶1
2
= 0.79
• No growth in the steady state: only transitional dynamics.
21

The Neoclassical Growth Model and Growth
• We can take the absence of growth as a positive lesson.
• Illuminates why capital accumulation has an inherit limitation as a
source of economic growth:
1. Soviet Union.
2. Development theory of the 50’s and 60’s.
3. East Asian countries today?
• Tells us we need to look some place else: technology.
22

Introducing Technological Progress
• Aggregate production function becomes
Y = Kα (AL)1−α
• A : Level of technology in period t.
• Key assumption: constant positive rate of technological progress:
˙A
A
= g > 0
• Growth is exogenous.
23

Balanced Growth Path
• Situation in which output per worker, capital per worker, and con-
sumption per worker grow at constant (but potentially diﬀerent) rates
• Steady state is just a balanced growth path with zero growth rate
• For Neoclassical Growth model, in BGP: gy = gk = gc
24

Proof
• Capital Accumulation Equation ˙K = sY − δK
• Dividing both sides by K yields gK ≡
˙K
K = sY
K − δ
• Remember that gk ≡
˙k
k =
˙K
K − n
• Hence
gk ≡
˙k
k
= s
Y
K
− (n + δ)
• In BGP gk constant. Hence Y
K constant. It follows that gY = gK.
Therefore gy = gk
25

What is the Growth Rate?
• Output per worker
y =
Y
L
=
Kα (AL)1−α
L
=
Kα
Lα
(AL)1−α
L1−α
= kαA1−α
• Take logs and diﬀerentiate gy = αgk + (1 − α)gA
• We proved gk = gy and we use gA = g to get
gk = αgk + (1 − α)g = g = gy
• BGP growth rate equals rate of technological progress. No TP, no
growth in the economy.
26

Analysis of Extended Model
• In BGP variables grow at rate g. Want to work with variables that are
constant in long run. Deﬁne:
˜y =
y
A
=
Y
AL
˜k =
k
A
=
K
AL
• Repeat the analysis with new variables:
˜y = ˜kα
˙˜k = s˜y − (n + g + δ)˜k
˙˜k = s˜kα − (n + g + δ)˜k
27

Close-Form Solution
• Repeating all the steps than in the basic model we get:
˜k(t) =
³
s
δ+n+g +
³
˜k1−α
0 − s
δ+n+g
´
e−λt
´ 1
1−α
˜y(t) =
³
s
δ+n+g +
³
˜k1−α
0 − s
δ+n+g
´
e−λt
´ α
1−α
• Interpretation.
28

Balanced Growth Path Analysis I
• Solve for ˜k∗ analytically
0 = s˜k∗α
− (n + g + δ)˜k∗
˜k∗ =
Ã
s
n + g + δ
! 1
1−α
• Therefore
˜y∗ =
Ã
s
n + g + δ
! α
1−α
29

Balanced Growth Path Analysis II
k(t) = A(t)
Ã
s
n + g + δ
! 1
1−α
y(t) = A(t)
Ã
s
n + g + δ
! α
1−α
K(t) = L(t)A(t)
Ã
s
n + g + δ
! 1
1−α
Y (t) = L(t)A(t)
Ã
s
n + g + δ
! α
1−α
30

Evaluation of the Model: Growth Facts
1. Output and capital per worker grow at the same constant, positive
rate in BGP of model. In long run model reaches BGP.
2. Capital-output ratio K
Y constant along BGP
3. Interest rate constant in balanced growth path
4. Capital share equals α, labor share equals 1 − α in the model (always,
not only along BGP)
5. Success of the model along these dimensions, but source of growth,
technological progress, is left unexplained.
31

Evaluation of the Model: Development Facts
1. Differences in income levels across countries explained in the model
by differences in s, n and δ.
2. Variation in growth rates: in the model permanent differences can only
be due to differences in rate of technological progress g. Temporary
differences can be explained by transition dynamics.
3. That growth rates are not constant over time for a given country can
be explained by transition dynamics and/or shocks to n, s and δ.
4. Changes in relative position: in the model countries whose s moves up,
relative to other countries, move up in income distribution. Reverse
with n.
32

Interest Rates and the Capital Share
• Output produced by price-taking ﬁrms
• Hire workers L for wage w and rent capital Kfrom households for r
• Normalization of price of output to 1.
• Real interest rate equals r − δ
33

Proﬁt Maximization of Firms
max
K,L
Kα (AL)1−α
− wL − rK
• First order condition with respect to capital K
αKα−1 (AL)1−α
− r = 0
α
µ
K
AL
¶α−1
= r
α˜kα−1 = r
• In balanced growth path ˜k = ˜k∗, constant over time. Hence in BGP
rconstant over time, hence r − δ (real interest rate) constant over
time.
34

Capital Share
• Total income = Y, total capital income = rK
• Capital share
capital share =
rK
Y
=
αKα−1 (AL)1−α K
Kα (AL)1−α
= α
• Labor share = 1 − α.
35

Wages
• First order condition with respect to labor L
(1 − α)Kα(LA)−αA = w
(1 − α)˜kαA = w
• Along BGP ˜k = ˜k∗, constant over time. Since Ais growing at rate g,
the wage is growing at rate g along a BGP.
36

Human Capital and Growth
1

Introduction to Human Capital
• Education levels are very diﬀerent across countries.
• Rich countries tend to have higher educational levels than poor coun-
tries.
• We have the intuition that education (learning skills) is an important
factor in economic growth.
2

Production Function
• Cobb-Douglas aggregate production function:
Y = KαHβ (AL)1−α−β
• Again we have constant returns to scale.
• Human capital and labor enter with a diﬀerent coeﬃcient.
3

Inputs Accumulation
• Society accumulates human capital according to:
˙H = shY − δH
• Capital accumulation equation:
˙K = skY − δK
• Technology progress:
˙A
A = g > 0.
• Labor force grows at constant rate:
˙L
L = n > 0.
4

Rewriting the Model in Efficiency Units
• Redefine the variables in efficiency units:
˜x ≡
X
AL
• Then, diving the production function by AL:
˜y = ˜kα˜hβ
• Decreasing returns to scale in per efficiency units.
5

Human Capital Accumulation
• The evolution of inputs is determined by:
˙˜k = sk
˜kα˜hβ − (n + g + δ)˜k
˙˜h = sh
˜kα˜hβ − (n + g + δ)˜h
• System of two diﬀerential equations.
• Solving it analytically it is bit tricky so we will only look at the BGP.
6

Phase Diagram
• Solving the system analytically it is bit tricky.
• Alternatives:
1. Use numerical methods.
2. Linearize the system.
3. Phase diagram.
7

k
h
PHASE DIAGRAM
˙h
·
k
SOLOW MODEL WITH HUMAN CAPITAL

Balanced Growth Path Analysis I
• To ﬁnd the BGP equate both equations to zero:
sk
˜k∗α˜h∗β − (n + g + δ)˜k∗ = 0
sh
˜k∗α˜h∗β − (n + g + δ)˜h∗ = 0
• From ﬁrst equation:
˜h∗ =
Ã
(n + g + δ)
sk
˜k∗1−α
!1
β
8

Balanced Growth Path Analysis II
• Plugging it in the second equation
sh
˜k∗α(n + g + δ)
sk
˜k∗1−α − (n + g + δ)
Ã
n + g + δ
sk
˜k∗1−α
!1
β
= 0 ⇒
sh
sk
˜k∗ =
Ã
n + g + δ
sk
˜k∗1−α
!1
β
• Work with the expression.
9

Some Algebra
sh
sk
˜k∗ =
Ã
n + g + δ
sk
˜k∗1−α
!1
β
⇒
˜k
∗1−1−α
β = ˜k
∗−1−α−β
β =
sk
sh
Ã
(n + g + δ
sk
!1
β
⇒
˜k∗ =
Ã
s1−β
k sβ
h
n+g+δ
! 1
1−α−β
˜h∗ =
Ã
sα
k s1−α
h
n+g+δ
! 1
1−α−β
10

Evaluating the Model I
• Using the production function:
˜y =
Y
AL
= ˜kα˜hβ =


s
1−β
k s
β
h
n + g + δ


α
1−α−β


sα
k s1−α
h
n + g + δ


β
1−α−β
⇒
y =
Y
L
=


s
1−β
k s
β
h
n + g + δ


α
1−α−β


sα
k s1−α
h
n + g + δ


β
1−α−β
A
• Given some initial value of technology A0 we have:
y =


s
1−β
k s
β
h
n + g + δ


α
1−α−β


sα
k s1−α
h
n + g + δ


β
1−α−β
A0egt
11

Evaluating the Model II
• Taking logs:
log y = log A0 + gt −
α + β
1 − α − β
log (n + g + δ) +
+
α
1 − α − β
log sk +
β
1 − α − β
log sh
• What if we have a lot of countries i = 1, ..., n?
• We can assume that log A0 = a + εi
• Also assume that g and δ are constant across countries.
12

Evaluating the Model III
• Then we have:
log yi = a + gt −
α + β
1 − α − β
log (ni + g + δ) +
+
α
1 − α − β
log ski +
β
1 − α − β
log shi + εi
• This is a functional form that can be taken to the data.
13

47
TFPgrowthrate,1965-95
(Laborshare=0.65,7%returntoeducation)
Figure 1: Relation of TFP growth to schooling rate
Human capital investment rate, 1965-95
0 .05 .1 .15
-.04
-.02
0
.02
.04
TZA
NER
RWA
MOZ
MWI
MLIUGA
CAF
PNGBEN
SEN
CMR
ZAR
GTM
ZMB
PAK
BGD
KEN
TGO
NPLHNDSLV
ZWE
PRY
THA
BRABWA
IDN
IND
BOL
GHA
TUN
DOM
VEN
TUR
NIC
DZAZAF
CRI
COL
MUS
ARG
ITA
PRT
HKG
ECU
MYS
URY
MEXCHE
CHL
GRCSWE
COG
SGP
GBR
LKA
PER
SYR
EGY
AUT
PAN
FRAESP
BEL
AUS
JPNNOR
TTO
CAN
ISR
DNK
USA
KOR
FIN
PHL
NLD
JAM
NZL
IRL
JOR
TFPgrowthrate,1965-95
(Laborshare=0.65,7%returntoeducation)
Figure 6: Relation of TFP growth to labor force growth rate
Labor force growth rate, 1965-95
0 .02 .04 .06
-.04
-.02
0
.02
.04
GBRSWE
BELDNK
PRT
FIN
ITA
URY
AUT
NOR
FRAGRC
CHE
JPN
ESPNLD
IRL
USA
ARG
NZL
TTO
MOZ
AUS
CAN
JAM
CAF
MLI
LKA
CHLMUS
RWA
NPL
IND
BOL
BGD
PNG
IDN
HKG
KOR
CMR
SLV
COG
ZAFEGY
GHA
GTM
ISR
BEN
BRA
SEN
UGATUR
TUN
PER
ZAR
SGP
NER
PAN
TGO
COL
MWI
ZMB
PHL
PAK
TZA
DOM
THA
MEX
MYS
ECU
NIC
PRY
HND
DZA
ZWE
VENCRI
KEN
SYR
BWA
JOR

Convergence
and
World Income Distribution
1

The Convergence Hypothesis
• Fact: Enormous variation in incomes per worker across countries
• Question: Do poor countries eventually catch up?
• Convergence hypothesis: They do, in the right sense!
• Main prediction of convergence hypothesis: Poor countries should
grow faster than rich countries.
• Let us look at the data.
2

Neoclassical Growth Model and Convergence
Countries with same s, n, δ, α, g
• eventually same growth rate of output per worker and same level of
output per worker (absolute convergence).
• countries starting further below the balanced growth path (poorer
countries) should grow faster than countires closer to balanced growth
path.
• seems to be the case for the sample of now industrialized countries.
3

Countries with same g, but potentially diﬀering s, n, δ, α
• countries have diﬀerent balanced growth path.
• countries that start further below their balanced growth path (coun-
tires that are poor relative to their BGP) should grow faster than rich
countries (relative to their BGP). This is called conditional conver-
gence.
• data for full sample lend support to conditional convergence.
4

World Income Distribution
What is happening with the distribution of world income?
Look at the data again.
5

Conclusion: The Neoclassical Growth Model
• Oﬀers a simple and elegant account of a number of growth facts.
• However:
1. leaves unexplained factors that make countries leave (or not attain)
their BGP.
2. leaves unexplained why certain countries have higher s, n than oth-
ers.
3. leaves unexplained technological progress, the source of growth.
6

Figure 1.a: Growth Rate Versus Initial Per Capita GDP
Per Capita GDP, 1885
GrowthRateofPerCapitaGDP,1885−1994
0 1000 2000 3000 4000 5000
1
1.5
2
2.5
3
JPN
FIN
NOR
ITL
SWE
CAN
FRA
DNK
AUT
GER
BEL
USA
NLD
NZL
GBR
AUS

Figure 1.b: Growth Rate Versus Initial Per Capita GDP
Per Worker GDP, 1960
0 0.5 1 1.5 2 2.5
x 10
4
0
1
2
3
4
5
TUR
POR
JPN
GRC
ESP
IRL
AUT
ITL
FIN
FRA
GER
BEL
NOR
GBR
DNK
NLD
SWE
AUS
CAN
CHE
NZL
USA

Figure 1.c: Growth Rate Versus Initial Per Capita GDP
Per Worker GDP, 1960
0 0.5 1 1.5 2 2.5
x 10
4
−4
−2
0
2
4
6
LUX
USA
CAN
CHE
BEL
NLD
ITA
FRA
AUS
GER
NOR
SWE
FIN
GBR
AUT
ESP
NZL
ISL
DNK
SGP
IRL
ISR
HKG
JPN
TTO
OAN
CYP
GRC
VEN
MEX
PRT
KOR
SYRJOR
MYS
DZA
CHL
URY
FJI
IRN
BRA
MUS
COL
YUG
CRI
ZAF
NAM
SYC
ECU
TUN
TUR
GAB PANCSK
GTM DOM
EGY
PER
MAR
THA
PRY
LKA
SLV
BOL
JAM
IDN
BGD
PHL
PAK
COG
HND
NIC
IND CIV
PNG
GUY
CIV
CMR
ZWESEN
CHN
NGA
LSO
ZMB
BEN
GHA
KEN
GMB
MRT
GIN
TGO
MDG
MOZ RWA
GNB COM
CAF
MWI
TCD
UGA
MLI
BDI
BFA
LSO
MLI
BFA
MOZ
CAF

Figure 3.a: Population-Weighted Variance of Log Per Capita Income:
125 Countries
0.70
0.80
0.90
1.00
1.10
1.20
1.30
1.40
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
Unweighted SD Weighted SD

Figure3b:Estimated World Income Distributions (Various Years)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
4 5 6 7 8 9 10 11 12
log(income)
1970 1980 1990 1998

Figure 3b1: Individual-Country and Global Distributions: 1970
0
30,000
60,000
90,000
120,000
150,000
180,000
4 5 6 7 8 9 10 11 12 13
log(income)
individual Countries Global
0
30,000
60,000
90,000
120,000
150,000
180,000
210,000
4 5 6 7 8 9 10 11 12 13
log(income)individual Countries Global

0
30,000
60,000
90,000
120,000
150,000
180,000
210,000
240,000
4 5 6 7 8 9 10 11 12 13
log(income)
individual Countries Global
0
50,000
100,000
150,000
200,000
250,000
4 5 6 7 8 9 10 11 12 13
log(income)
Individual Countries Global

Figure 3b5: Income Distribution: China
0
20,000
40,000
60,000
80,000
100,000
5 6 7 8 9 10
log(income)
1970 1980 1990 1998
Figure 3b6: Income Distribution: India
0
20,000
40,000
60,000
80,000
100,000
5 6 7 8 9 10
1970 1980 1990 1998
Figure 3b7: Income Distribution: USA
0
5,000
10,000
15,000
20,000
5 6 7 8 9 10 11 12 13
log(income)
1970 1980 1990 1998

Figure 3b7: Income Distribution: Indonesia
0
3,000
6,000
9,000
12,000
15,000
18,000
5 6 7 8 9 10
log(income)
1970 1980 1990 1998
Figure 3b8: Income Distribution: Brazil
0
2,000
4,000
6,000
8,000
5 6 7 8 9 10 11
log(income)
1970 1980 1990 1998
Figure 3b9: Income Distribution: Pakistan
0
3,000
6,000
9,000
12,000
5 6 7 8 9 10
log(income)
1970 1980 1990 1998

Figure 3b10: Income Distribution: Japan
0
3,000
6,000
9,000
12,000
5 6 7 8 9 10 11 12 13
log(income)
1970 1980 1990 1998
Figure 3b12: Income Distribution: Nigeria
0
1,000
2,000
3,000
4,000
5,000
6,000
7,000
4 5 6 7 8 9 10
log(income)
1970 1980 1990 1998
Figure 3b11: Income Distribution: Bangladesh
0
3000
6000
9000
12000
5 6 7 8 9 10
log(income)
1970 1980 1990 1998

Figure 3.c: Poverty Rates
0
0.08
0.16
0.24
0.32
0.4
0.48
1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998
Less than $1 a Day Less than 2$ a Day

Figure 3.d: Poverty Headcount (in millions of people)
0
200
400
600
800
1000
1200
1400
1600
1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998
Less than $1 a Day Less than $2 a Day

Figure 3.f: Poverty Headcounts for World Regions: 1$/Day
0
100
200
300
400
500
600
1970 1980 1990 1998
MillionsofPeople
World Africa Latin America Asia Asia Minus China China
Figure 3.e: Poverty Rates for World Regions: 1$/Day
0,00
0,10
0,20
0,30
0,40
0,50
1970 1980 1990 1998
FractionofWorldPopulation

Figure 3.g: Poverty Rates for World Regions: 2$/Day
0,00
0,10
0,20
0,30
0,40
0,50
0,60
0,70
0,80
1970 1980 1990 1998
FractionofWorldPopulation
Figure 3h: Poverty Headcounts for World Regions: 2$/Day
0
300
600
900
1.200
1.500
1970 1980 1990 1998
MillionsofPeople

Figure 4: Global Inequality: Variance of log-Income
0.80
0.90
1.00
1.10
1.20
1.30
1.40
1.50
1.60
1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998
Varlog Global Varlog Across-Country

Endogenous Growth Theory
1

New Growth Theory
• Remember from Solow Model:
g =
˙A
A
=
˙y
y
Growth depends on technological progress.
• Good thing: now we know where to look at.
• Challenge: we need a theory of technological progress.
• Paul Romer’s big contribution to economics.
2

Ideas as Engine of Growth
• Technology: the way inputs to the production process are transformed
into output.
• Without technological progress:
Y = KαL1−α
With technological progress:
Y = Kα (AL)1−α
• Technological progress due to new ideas: very diﬀerent examples.
• Why (and under what circumstances) are resources are spent on the
development of new ideas?
3

Ideas
• What is an idea?
• What are the basic characteristics of an idea?
1. Ideas are nonrivalrous goods.
2. Ideas are, at least partially, excludable.
4

Diﬀerent Types of Goods
1. Rivalrous goods that are excludable: almost all private consumption
goods, such as food, apparel, consumer durables fall into this group.
2. Rivalrous goods that have a low degree of excludability: tragedy of
the commons.
3. Nonrivalrous goods that are excludable: most of what we call ideas
falls under this point.
4. Nonrivalrous and nonexcludable goods: these goods are often called
public goods.
5

Nonrivalrousness and Excludability of Ideas
• Nonrivalrousness: implies that cost of providing the good to one more
consumer, the marginal cost of this good, is constant at zero. Pro-
duction process for ideas is usually characterized by substantial fixed
costs and low marginal costs. Think about software.
• Excludability: required so that firm can recover fixed costs of develop-
ment. Existence of intellectual property rights like patent or copyright
laws are crucial for the private development of new ideas.
6

Intellectual Property Rights and the Industrial Revolution
• Ideas engine of growth.
• Intellectual property rights needed for development of ideas.
• Sustained growth recent phenomenon.
• Coincides with establishment of intellectual property rights.
7

Data on Ideas
• Measure technological progress directly through ideas
• Measure ideas via measuring patents
• Measure ideas indirectly by measuring resources devoted to develop-
ment of ideas
8

Important Facts from Data
• Number of patents issued has increased: in 1880 roughly 13,000
patents issued in the US, in 1999 150,000
• More and more patents issued in the US are issued to foreigners. The
number of patents issued to US ﬁrms or individuals constant at 40,000
per year between 1915 and 1991.
• Number of researchers engaged in research and development (R&D)
in the US increased from 200,000 in 1950 to 1,000,000 in 1990.
• Fraction of the labor force in R&D increased from 0.25% in 1950 to
0.75% in 1990.
9

Infrastructure or Institutions
• Question: why does investment rate s differ across countries?
• Answer: some countries have political institutions that make investing
more profitable than others.
• Investment has costs and benefits: some countries invest more than
others because either the costs of investment are lower or the benefits
are higher.
10

Cost of Investment
• Cost of investment: resources to develop idea, purchase of buildings
and equipment.
• Cost of obtaining all legal permissions.
• Hernando de Soto “The Other Path” (1989).
• Deﬁcient or corrupt bureaucracy can impede proﬁtable investment ac-
tivities.
11

Benefits of Investment
1. The size of the market. Depends on openness of the economy
2. The extent to which the benefits from the investment accrue to the
investor. Diversion of benefits due to high taxes, theft, corruption, the
need to bribe government officials or the payment of protection fees
to the Mafia or Mafia-like organizations.
3. Rapid changes in the economic environment in which firms and indi-
viduals operate: increase uncertainty of investors.
4. Data show that these considerations may be important
12

A Basic Model of Endogenous Growth
• Can we built a model that puts all this ideas together?
• Yes, Romer 1990. You can get a copy of the paper at the Class Web
Page.
• A bit of work but we can deal with it.
13

Basic Set-up of the Model I
• Model of Research and Growth.
• Three sectors: ﬁnal-goods sector, intermediate-goods sector and re-
search sector.
• Those that invent a new product and those that sell it do not need to
be the same: Holmes and Schmitz, Jr. (1990).
• Why? comparative advantage.
14

Basic Set-up of the Model II
• Total labor: L.
• Use for production of ﬁnal goods, LY , or to undertake research, LA =
L − LY .
• Total capital: K.
• Used for production of intermediate goods.
15

Final-Goods Sector I
• Competitive producers.
• Production Function:
Y = L1−α
Y
Z A
0
x (i)α di
• Optimization problem:
Π = L1−α
Y
Z A
0
x (i)α di − wY LY −
Z A
0
p (i) x (i) di
16

Final-Goods Sector II
• First Order Conditions:
wY = (1 − α) L−α
Y
Z A
0
x (i)α di = (1 − α)
Y
LY
p (i) = αL1−α
Y x (i)α−1
for ∀ i ∈ [0, A]
• Interpretation.
17

Intermediate-Goods Sector I
• Continuum of monopolist.
• Only use capital for production.
• Optimization problem:
π (i) = max
x(i)
p (i) x (i) − rx (i)
• Since p (i) = αL1−α
Y x (i)α−1 we have:
π (i) = max
x(i)
αL1−α
Y x (i)α − rx (i)
18

Intermediate-Goods Sector II
• First Order Conditions:
α2L1−α
Y x (i)α−1 = r ⇒ αL1−α
Y x (i)α−1 =
1
α
r
p (i) =
1
α
r
• Interpretation: mark-up of a monopolist.
19

Intermediate-Goods Sector III
• Total demand:
p (i) =
1
α
r = αL1−α
Y x (i)α−1
⇒ x (i) =
µ
1
α2
r
¶ 1
α−1
LY
• The proﬁt of the monopolist:
π (i) =
1
α
rx (i) − rx (i) =
µ
1
α
− 1
¶
rx (i)
20

Aggregation I
• Solution of monopolist is independent of i:
x (i) = x and π (i) = π for ∀ i ∈ [0, A]
• Then:
Y = L1−α
Y
Z A
0
x (i)α di = L1−α
Y
Z A
0
xαdi = L1−α
Y xα
Z A
0
di = AxαL1−α
Y
21

Aggregation II
• Since the total amount of capital in the economy is given:
Z A
0
x (i) di = K
• Then:
Z A
0
x (i) di = x
Z A
0
di = Ax = K ⇒ x =
K
A
• Plugging it back:
Y = AxαL1−α
Y = A
µ
K
A
¶α
L1−α
Y = Kα (ALY )1−α
22

Aggregation III
• Taking logs and derivatives:
·
Y
Y
=
˙A
A
+ α
·
x
x
+ (1 − α)
·
LY
LY
• Then, in a balance growth path, since
·
LY
LY
= 0 and x =
³
1
α2r
´ 1
α−1
LY
are constant:
g =
·
Y
Y
=
˙A
A
23

Research Sector I
• Production function for ideas:
˙A = BALA
• Then:
g =
·
A
A
= BLA
24

Research Sector II
• PA is the price of the new design A.
• Arbitrage idea.
• By arbitrage:
rPA = π +
·
PA
25

Research Sector III
• Then r = π
PA
+
·
PA
PA
• In a BGP, r and
·
PA
PA
are constant and then also π
PA
. But since π is also
constant:
PA is constant ⇒
·
PA
PA
= 0
• And:
r =
π
PA
=
³
1
α − 1
´
rx
PA
⇒ PA =
µ
1
α
− 1
¶
x
26

Research Sector IV
• Each unit of labor in the research sector then gets:
wR = BAPA = BA
µ
1
α
− 1
¶
x
• Remember that the wage in the ﬁnal-goods sector was:
wY = (1 − α)
Y
LY
• By free entry into the research sector both wages must be equal:
w = wR = wY
27

Research Sector V
• Then:
BA
µ
1
α
− 1
¶
x = (1 − α)
Y
LY
• and with some algebra:
BA
α
=
Y
xLY
=
AxαL1−α
Y
xLY
= Axα−1L−α
Y ⇒
B
α
= xα−1L−α
Y
28

Balanced Growth Path I
• As in the Solow’s model:
·
K = sY − δK = sKα (ALY )1−α
− δK
• Dividing by K:
g =
·
K
K
= s
µ
K
A
¶α−1
L1−α
Y − δ = sxα−1L1−α
Y − δ
29

Balanced Growth Path II
• Then:
g = BLA = B(L − LY ) = s
B
α
LY − δ ⇒
BL + δ =
µ
1 +
s
α
¶
BLY ⇒
LY =
BL + δ
³
1 + s
α
´
B
=
1
1 + s
α
L +
δ
³
1 + s
α
´
B
• And we can compute all the remaining variables in the model.
30

Is the Level of R&D Optimal?
• Sources of ineﬃciency:
1. Monopoly power.
2. Externalities
• Possible remedies.
• Implications for Antitrust policy.
31

The Very Long Run
1

The Very Long Run
• Economist want to understand the growth experience of ALL human
history (Big History Movement).
• What are the big puzzles:
1. Why are there so big diﬀerences in income today?
2. Why did the West develop ﬁrst? or Why not China? or India?
• Data Considerations.
2

Where Does Data Come From?
• Statistics: Customs, Tax Collection, Census, Parish Records.
• Archeological Remains: Farms, Skeletons.
• Literary Sources: Memories, Diaries, Travel Books.
3

Some Basic Facts I
• For most human history, income per capita growth was glacially slow.
• Before 1500 little or no economic growth.
Paul Bairoch (Economics and World History: Myths and Paradoxes)
1. Living standards were roughly equivalent in Rome (1st century
A.D.), Arab Caliphates (10th Century), China (11th Century), In-
dia (17th century), Western Europe (early 18th century).
2. Cross-sectional diﬀerences in income were a factor of 1.5 or 2.
4

Some Basic Facts II
• Angus Maddison (The World Economy: A Millenial Perspective) cal-
culates 1500-1820 growth rates:
1. World GDP per capita: 0.05%.
2. Europe GDP per capita: 0.14%.
• After 1820: great divergence in income per capita.
5

Europe Becomes Dominant
• From 1492 to 1770, diﬀerent human populations come into contact.
European countries expand until early 20th century:
1. American and Australia: previous cultures were nearly wiped out.
2. Asia: partial control.
3. Africa: somehow in the middle.
• Proximate causes: weaponry and social organization of Europeans was
more complex.
• Ultimate causes: why?
6

Possible Explanations
• Geography.
• Colonies.
• Culture.
7

Geography
• How can Geography be important?
• Examples:
1. Europe is 1/8 of the size of Africa but coastline is 50% longer.
2. Wheat versus Rice, Braudel (The Structures of Everyday Life: Civ-
ilization and Capitalism, 15th-18th Century).
• Let’s look at a map.
8

Jared Diamond (Guns, Germs, and Steel): geography.
• Euroasia is bigger (50% than America, 250% as Sub-saharian Africa,
800% than Australia):
1. More plants i.e. out of 56 food grains, 39 are native to Euroasia,
11 to America, 4 to Sub-saharian Africa, and 2 to Australia.
2. More animals to domesticate: cows/pigs/horses/sheeps/goats ver-
sus llamas and alpacas.
• Euroasia is horizontal: transmission of technology, plants, and animals.
• Consequence: higher population density→guns and germs.
9

Why Not China?
• But, how can Diamond explain China?
• Between the 8th and the 12th century, China experienced a burst of
economic activity: gunpowder, printing, water-powered spinning wheel
• Voyages of exploration by admiral Zheng: Louise Levathes (When
China Ruled the Seas: The Treasure Fleet of the Dragon Throne,
1405-1433).
• With the arrival of the Ming dynasty (1368), China stagnates.
• Europe gets ahead.
10

http://www.kungree.com/bib/focus/ship.gif
http://www.kungree.com/bib/focus/ship.gif [1/31/2005 7:36:26 PM]

Eric Jones (The European Miracle): geography, hypothesis 1.
• China was first unified around 221 B.C. Since then, except for relatively
short periods, unified state (last partition ended with arrival of Mongols
in 13th century).
• Europe has never been unified since the fall of Roman Empire (476
a.d.).
• Why? Dispersion of core areas.
11

Kenneth Pomeranz (The Great Divergence: China, Europe, and the Mak-
ing of the Modern World Economy): geography, hypothesis 2.
• Coal:
1. Far away from production centers
2. Steam engine versus ventilations.
• Environmental limits.
12

Colonies
Immanuel Wallerstein (The Modern World System).
• Small initial diﬀerences in income.
• Patterns of labor control and trade policies created “plantation” economies.
• Trade: primary goods for manufacturing.
• Forward and Backward linkages.
13

Differences across Colonies
Daron Acemoglu, Simon Johnson, and James Robinson (The Colonial Ori-
gins of Comparative Development: An Empirical Investigation).
• Differences in settlers mortality.
• Differences in outcomes:
1. British America: 9 universities for 2.5 million people.
2. Spanish and Portuguese America: 2 universities for 17 million peo-
ple.
14

Cultural Diﬀerences: Yes
• Max Weber (The Protestant Ethic and the Spirit of Capitalism)
• Letter from the Chinese emperor Qian Long to King George III of
England:
“Our dynasty’s majestic virtue has penetrated unto every country un-
der Heaven...As your Ambassador can see for himself, we possess all
things. I set no value on objects strange or ingenious, and have no use
for your country’s manufactures”.
• Leibniz’s Instructions to a European traveler to China:
“Not too worry so much about getting things European to the Chinese,
but rather about getting remarkable Chinese inventions to us”.
15

Cultural Differences: No
• A Western traveler, 1881
“The Japanese are a happy race, and being content with little, are not
likely to achieve much”.
• Karen Kupperman (Providence Island, 1630-1641 : The Other Puritan
Colony):
Documents differences between Providence Island and New England.
• Philip Benedict (The Faith and Fortunes of France’s Huguenots):
Differences between Catholics and Huguenots in France.
16

An Empirical Application:
Population Growth and Technological Change since 1 Million B.C.
• Basic lesson so far: growth depends on technology progress.
• Intuition: more people probably must imply higher knowledge accu-
mulation.
• Growth and population may be closely link.
• Empirical evidence.
17

A Simple Model
• Production function:
Y = Tα (AL)1−α
• Technology progress:
˙A = BAL
• Malthusian assumption:
Y
L
= y∗
18

Solving the Model
• We ﬁnd the level of population allowed by a technology:
Tα (AL)1−α
L
= y∗ ⇒ L∗ =
Ã
1
y∗
!1
α
A
1−α
α T
• Growth rate of population:
˙L∗
L∗
=
1 − α
α
˙A
A
• Then:
nt =
˙L∗
L∗
=
1 − α
α
BAL
A
=
1 − α
α
BL
19

Time Series Evidence
• A first look at the data.
• Regression:
nt = −0.0026
(0.0355)
+ 0.524
(0.0258)
Lt
R2 = 0.92, D.W = 1.10
• Robust to different data sets and specifications.
20

Cross-Section Evidence
• World population was separated from 10,000 BC to circa 1500 AD
• Population and Population Density circa 1500:
Land Area Population Pop/km2
“Old World” 83.98 407 4.85
Americas 38.43 14 0.36
Australia 7.69 0.2 0.026
Tasmania 0.068 0.0012-0.005 0.018-0.074
Flinders Islands 0.0068 0.0 0.0
• England vs. Europe and Japan vs. Asia.
21

-0,005
0
0,005
0,01
0,015
0,02
0,025
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Population
PopulationGrowthRate

Introduction to General Equilibrium I:
Households
1

The Representative Household
• Who is the representative household?
• Robinson Crusoe in a desert island.
• Justiﬁcation: aggregation.
• When does aggregation work and when does it not?
2

What are we going to do?
• Think about the goods existing in the economy.
• Think about what does Robinson prefer.
• Think about his constraints.
• Think about what will Robinson do given his preferences and his con-
straints
3

Commodity Space
• 2 goods, consumption c and leisure 1 − l.
• Each goods set:
1. c ∈ <+
2. l ∈ [0, 1]
• Then (c, l) ∈ <+ × [0, 1].
4

Preferences
• Preferences: binary relation º deﬁned over pairs (c, l):
(ci, li) º
³
cj, lj
´
• Assumptions on preferences:
1. Complete: for ∀ (ci, li) ,
³
cj, lj
´
∈ <+ × [0, 1] either (ci, li) º
³
cj, lj
´
or
³
cj, lj
´
º (ci, li).
2. Reﬂexive: for ∀ (ci, li) ∈ <+ × [0, 1] (ci, li) º (ci, li).
3. Transitive: for ∀ (ci, li) ,
³
cj, lj
´
, (ck, lk) ∈ <+ × [0, 1], if (ci, li) º
³
cj, lj
´
and
³
cj, lj
´
º (ck, lk) ⇒ (ci, li) º (ck, lk).
5

Indiﬀerence Curves
• Loci of pairs such that:
(ci, li) º
³
cj, lj
´
³
cj, lj
´
º (ci, li)
• If we assume that preferences are strictly monotonic, convex and nor-
mal, the indiﬀerence curves are:
1. Negative sloped in 1 − l.
2. Convex.
6

Utility Function I
• Working directly with binary relations diﬃcult.
• Can we transform them into a function?
• Why is this useful?
7

Utility Function II
• Definition: a real-value function u : <2 → < is called a utility func-
tion representing the binary relation º defined over pairs (c, l) if for
∀ (ci, li) ,
³
cj, lj
´
∈ <+ × [0, 1], (ci, li) º
³
cj, lj
´
⇔ u (ci, li) ≥
u
³
cj, lj
´
.
• Theorem: if the binary relation º is complete, reflexive, transitive,
strictly monotone and continuous, there exist a continuous real-value
function u that represents º .
• Proof (Debreu, 1954): intuition.
8

Utility Function III
• Utility function and monotone transformations.
• Interpretation of u.
• Diﬀerentiability of u.
9

Budget Constraint
• Leisure 1 − l ⇒labor supply l.
• Wage w.
• Then
c = lw
• Interpretation for Robinson.
10

Household’s Problem
• Problem for Robinson is then
max
c,l
u (c, 1 − l)
s.t. c = lw
• First order condition:
−
ul
uc
= w
• Interpretation: marginal rate of substitution equal to relative price of
leisure.
11

A Parametric Example
• u (c, l) = log c + γ log (1 − l)
• FOC+Budget constraint:
γ
c∗
1 − l∗
= w
c∗ = l∗w
• Then:
l∗ =
1
1 + γ
12

Income and Substitution Effect
• We will follow the Hicksian decomposition.
• Substitution Effect: changes in w make leisure change its relative price
with total utility constant.
• Income Effect: changes in w induce changes in total income even if l∗
stays constant.
• For u (c, l) = log c+γ log (1 − l) income and substitution effect cancel
each other!
13

Theory and Data
• Can we use the theory to account for the data?
• What are the trends in labor supply?
14

*The compensation series is an index of hourly compensation
in the business sector, deflated by the consumer price index
for all urban consumers.
Sources: Tables 1 and 14
Chart 1
Two Aggregate Facts
Average Weekly Hours Worked per Person
and Real Compensation per Hour Worked*
in the United States, 1950–90
Index
(1982 = 100) Hours
110
100
90
80
70
60
50
40
30
20
10
0
1950 1960 1970 1980 1990
Compensation
per Hour
Hours
per Person

50
40
30
20
10
0
10 20 30 40 50 60 70 80
Hours
Age (in Years)
Charts 2–4
Possible Shifts in Hours Worked
Extrapolated Average Weekly Hours Worked per Person
by Cohorts at Various Ages in the United States
Chart 3 Females
Year Born
1866–75
1876–85
1886–95
1896–1905
1906–15
1916–25
1926–35
1936–45
1946–55
1956–65
1966–75
50
40
30
20
10
0
10 20 30 40 50 60 70 80
Hours
Age (in Years)
Chart 2 Males

Chart 4 Total Population
50
40
30
20
10
0
10 20 30 40 50 60 70 80
Hours
Age (in Years)Sources: Tables 8–10
Year Born
1866–75
1876–85
1886–95
1896–1905
1906–15
1916–25
1926–35
1936–45
1946–55
1956–65
1966–75

Table 1
A Look Behind an Aggregate Fact
In the United States, 1950–90
Average Weekly Hours Worked
Employment-to-
Year Per Person Per Worker Population Ratio
1950 22.03 40.71 .52
1960 20.97 37.83 .52
1970 20.55 36.37 .53
1980 22.00 35.97 .58
1990 23.62 36.64 .61
% Change
1950–90 7.2 –10.0 17.3
Source: U.S. Department of Commerce, Bureau of the Census

Table 3 By Age
Weekly Hours Worked per Person by Age (in Years)
Year 15–24 25–34 35–44 45–54 55–64 65–74 75–84
1950 17.47 24.92 27.09 26.31 22.19 12.03 3.93
1960 14.15 24.73 27.00 27.63 22.58 8.43 2.97
1970 14.05 26.16 28.03 28.27 23.28 6.91 2.17
1980 19.64 28.80 29.89 28.16 20.68 5.11 1.39
1990 19.13 30.83 32.62 31.47 20.75 5.15 1.18
% Change
1950–90 9.5 23.7 20.4 19.6 –6.5 –57.2 –70.0
Tables 2–4
A Distribution of Hours Worked
Average Weekly Hours Worked per Person
for Demographic Categories in the United States, 1950–90
Table 2 By Sex
Weekly Hours Worked per Person by
Sex
Total
Year Population Males Females
1950 22.03 33.46 10.95
1960 20.97 30.70 11.82
1970 20.55 28.54 13.29
1980 22.00 28.30 16.24
1990 23.62 28.53 19.09
% Change
1950–90 7.2 –14.7 74.3
Table 4 By Marital Status*
Weekly Hours Worked per Person by Marital Status
Married With Spouse
Year Present Absent Single Widowed Divorced
1950 23.89 23.11 28.10 11.82 28.65
1960 23.86 20.43 25.72 10.37 26.31
1970 24.31 20.50 24.19 9.41 26.17
1980 24.15 22.71 25.42 6.86 27.22
1990 26.26 22.22 27.73 5.98 28.41
% Change
1950–90 9.9 –3.9 –1.3 –49.4 –.8
*This excludes individuals less than 25 years old.

Spouse Total
Present
Spouse Total
Absent
Youngest Child
Under
6 Years Old
Youngest Child
6–17 Years Old
Tables 5–6
A More Comprehensive Distribution of Hours Worked
Average Weekly Hours Worked per Person for Sets of Demographic Categories in the United States, 1950–90
Table 5 Married . . .
Status Sex Year 15–24 25–34 35–44 45–54 55–64 65–74 75–84
Males 1950 38.69 41.14 43.06 41.95 37.58 23.39 9.77
1960 36.58 40.67 41.79 40.99 35.74 14.74 6.22
1970 34.19 40.30 41.52 40.65 34.74 11.51 4.08
1980 33.63 38.70 40.22 38.89 29.83 8.20 2.95
1990 34.18 40.25 41.34 40.03 28.39 7.71 2.50
% Change 1950–90 –11.7 –2.2 –4.0 –4.6 –24.5 –67.0 –74.4
Females 1950 9.17 8.09 9.60 8.61 4.60 1.79 .56
1960 10.00 9.10 12.35 13.55 8.66 2.27 .94
1970 14.65 12.21 14.95 16.18 11.75 2.55 1.03
1980 18.36 18.77 19.64 18.16 11.95 2.48 .70
1990 21.13 23.90 25.41 24.04 13.83 2.79 .65
% Change 1950–90 130.4 195.4 164.7 179.2 200.7 55.9 16.1
Females 1950 3.40 4.60 6.49 6.41 4.24 3.93 6.82
1960 5.71 5.75 6.36 9.17 7.25 2.10 2.25
1970 9.08 8.33 9.04 12.11 10.13 3.70 6.02
1980 11.72 13.47 13.00 11.77 9.32 1.34 .30
1990 15.49 19.48 19.62 18.55 13.11 6.61 7.86
% Change 1950–90 355.6 323.5 202.3 189.4 209.2 68.2 15.2
Females 1950 3.89 5.57 7.64 6.81 4.50 2.28 10.08
1960 13.27 13.44 13.75 11.99 8.75 2.53 1.45
1970 16.23 15.90 15.85 14.49 11.41 4.15 6.81
1980 15.46 20.79 20.01 16.76 11.91 3.90 3.41
1990 23.43 24.85 25.70 23.01 15.09 5.98 11.01
% Change 1950–90 502.3 346.1 236.4 237.9 235.3 162.3 9.2
Males 1950 24.17 27.54 31.56 30.48 26.62 16.54 5.93
1960 17.13 25.80 27.83 29.49 24.69 9.66 3.47
1970 16.49 27.12 29.67 30.48 25.06 9.13 2.94
1980 25.27 30.64 31.99 29.18 20.58 5.99 2.40
1990 21.03 27.31 28.80 29.84 21.63 6.23 1.48
% Change 1950–90 –13.0 –.8 –8.7 –2.1 –18.7 –62.3 –75.0
Females 1950 15.37 20.00 22.26 19.74 13.82 4.42 1.04
1960 14.24 17.52 20.51 20.74 15.58 4.30 1.57
1970 16.05 18.03 20.17 21.43 17.25 5.43 2.16
1980 17.12 21.77 22.78 21.32 15.79 3.75 1.50
1990 15.89 21.95 25.26 24.22 15.72 4.01 .84
% Change 1950–90 3.4 9.8 13.5 22.7 13.7 –9.3 –19.2

Single
Widowed
Divorced
Table 6 . . . And Not Married
Status Sex Year 15–24 25–34 35–44 45–54 55–64 65–74 75–84
Males 1950 18.29 31.58 33.82 31.97 27.18 15.47 6.12
1960 12.67 30.61 30.35 28.98 24.30 9.74 5.01
1970 11.37 29.78 29.82 28.03 22.60 8.58 4.17
1980 19.23 30.55 29.01 26.24 19.60 6.26 2.06
1990 18.76 31.50 30.17 26.64 17.87 5.83 2.03
% Change 1950–90 2.6 –.3 –10.8 –16.7 –34.3 –62.3 –66.8
Females 1950 14.33 30.58 30.51 28.61 22.77 10.36 3.14
1960 10.70 29.33 29.37 28.94 24.40 10.63 3.35
1970 10.43 28.82 27.65 27.62 24.23 8.41 3.07
1980 17.23 29.15 28.24 25.76 20.68 4.93 1.19
1990 17.35 29.73 30.21 27.59 18.55 4.98 1.02
% Change 1950–90 21.1 –2.8 –1.0 –3.6 –18.5 –51.9 –67.5
Males 1950 19.65 33.50 35.76 34.12 29.15 14.99 4.67
1960 19.74 32.00 31.33 31.97 25.95 9.24 3.56
1970 19.68 29.63 32.08 31.93 25.36 7.24 2.34
1980 18.64 28.31 29.66 29.10 20.89 5.24 1.70
1990 15.20 26.62 28.70 29.06 18.32 4.90 1.38
% Change 1950–90 –22.6 –20.5 –19.7 –14.8 –37.2 –67.3 –70.4
Females 1950 17.02 21.75 23.90 20.11 12.96 4.31 .83
1960 15.64 17.61 22.82 23.35 15.71 4.72 1.18
1970 17.66 21.00 21.85 23.52 17.82 4.20 1.04
1980 17.12 17.25 21.13 20.71 15.68 3.30 .66
1990 10.56 18.50 24.06 24.41 15.16 3.59 .57
% Change 1950–90 –38.0 –14.9 .7 21.4 17.0 –16.7 –31.3
Males 1950 29.53 32.82 34.93 32.71 28.77 15.76 11.67
1960 24.08 30.54 31.51 29.50 25.75 9.75 4.95
1970 25.56 33.14 33.35 31.16 24.62 8.63 4.12
1980 29.16 33.73 34.39 30.90 22.34 6.09 2.65
1990 29.17 33.94 34.23 32.90 22.23 6.76 2.46
% Change 1950–90 –1.2 3.4 –2.0 .6 –22.7 –57.1 –78.9
Females 1950 25.27 28.72 30.68 27.32 21.99 10.07 1.96
1960 24.01 27.69 29.87 29.51 24.05 9.19 2.48
1970 25.16 27.59 29.73 29.71 25.04 7.99 3.26
1980 24.42 29.38 30.38 28.73 22.65 5.53 1.48
1990 23.26 29.13 32.82 31.86 23.73 6.68 1.49
% Change 1950–90 –8.0 1.4 7.0 16.6 7.9 –33.7 –24.0

Table 7
Partial Life-Cycle Proﬁles of Hours Worked by Males
Based on U.S. Census Data
Average Weekly Hours Worked per Person at Age (in Years)
Year Born 15–24 25–34 35–44 45–54 55–64 65–74 75–84
1866–75 — — — — — — 7.46
1876–85 — — — — — 20.75 5.12
1886–95 — — — — 35.34 13.31 3.53
1896–1905 — — — 40.06 33.60 10.65 2.57
1906–15 — — 41.41 39.15 32.84 7.71 2.16
1916–25 — 38.60 39.98 38.95 28.38 7.28 —
1926–35 22.65 38.20 39.79 37.20 26.73 — —
1936–45 17.65 37.89 38.59 37.75 — — —
1946–55 15.96 36.15 38.40 — — — —
1956–65 21.59 36.00 — — — — —
1966–75 20.23 — — — — — —

Table 8 By Males
Year Born 15–24 25–34 35–44 45–54 55–64 65–74 75–84
1866–75 34.08 41.71 45.23 42.55 39.27 25.10 7.46
1876–85 31.90 41.13 44.28 41.70 37.66 20.75 5.12
1886–95 29.67 40.50 43.20 41.01 35.34 13.31 3.53
1896–1905 27.92 39.93 42.35 40.06 33.60 10.65 2.57
1906–15 25.35 39.42 41.41 39.15 32.84 7.71 2.16
1916–25 23.00 38.60 39.98 38.95 28.38 7.28 1.17
1926–35 22.65 38.20 39.79 37.20 26.73 5.27 .39
1936–45 17.65 37.89 38.59 37.75 24.44 3.48 .00
1946–55 15.96 36.15 38.40 37.28 21.64 2.07 .00
1956–65 21.59 36.00 37.87 36.73 19.39 .33 .00
1966–75 20.23 35.27 37.23 36.57 16.95 .00 .00
Table 9 By Females
Year Born 15–24 25–34 35–44 45–54 55–64 65–74 75–84
1866–75 8.19 .08 3.63 4.44 4.65 4.12 1.01
1876–85 8.69 2.20 6.02 7.45 7.16 3.87 1.36
1886–95 9.19 4.61 8.19 10.18 8.85 4.23 1.29
1896–1905 10.00 6.43 10.38 12.58 12.30 3.98 .74
1906–15 10.18 8.58 13.17 16.48 14.71 3.13 .66
1916–25 10.68 11.84 14.70 18.38 13.91 3.48 .43
1926–35 12.43 11.87 16.97 19.78 15.41 3.23 .14
1936–45 10.73 15.03 21.53 25.48 16.45 2.98 .00
1946–55 12.18 21.63 26.96 28.48 17.03 2.93 .00
1956–65 17.68 25.67 31.05 31.85 18.06 2.75 .00
1966–75 17.99 30.27 35.74 35.87 18.95 2.59 .00
Tables 8–10
Extrapolated Life-Cycle Proﬁles of Hours Worked
U.S. Census Data Extrapolated as Explained in Appendix C*
*Highlighted areas indicate actual U.S. census data. The other data are extrapolations.

Table 10 By Total Population
Year Born 15–24 25–34 35–44 45–54 55–64 65–74 75–84
1866–75 20.91 20.40 24.36 23.90 22.50 14.34 3.93
1876–85 20.09 21.20 25.04 24.83 22.69 12.03 2.97
1886–95 19.26 22.13 25.53 25.69 22.19 8.43 2.17
1896–1905 18.81 22.78 26.16 26.31 22.58 6.91 1.39
1906–15 17.65 23.63 27.09 27.63 23.28 5.11 1.18
1916–25 16.75 24.92 27.00 28.27 20.68 5.15 .58
1926–35 17.47 24.73 28.03 28.16 20.75 4.06 .05
1936–45 14.15 26.16 29.89 31.47 20.14 3.11 .00
1946–55 14.05 28.80 32.62 32.75 19.09 2.44 .00
1956–65 19.64 30.83 34.49 34.24 18.56 1.53 .00
1966–75 19.13 32.86 36.65 36.27 17.84 .69 .00

Table 11
Lifetime Hours Worked
Average Weekly Hours Worked Between Ages 15 and 84
by Cohorts Born Between 1896 and 1945 in the United States
Weekly Hours Worked per Person by
Sex
Total
Year Born Population Males Females
1896–1905 17.85 28.15 8.06
1906–15 17.94 26.86 9.56
1916–25 17.62 25.34 10.49
1926–35 17.61 24.32 11.40
1936–45 17.85 22.83 13.17
% Change
1896–1945 0 –18.9 63.4
Sources: Tables 8–10

Table 12
Partial Life-Cycle Proﬁles for the Portion of the Population Employed . . .
Employment-to-Population Ratio at Age (in Years)
Sex Year Born 15–24 25–34 35–44 45–54 55–64 65–74 75–84
Males 1866–75 — — — — — — .19
1876–85 — — — — — .49 .15
1886–95 — — — — .80 .36 .11
1896–1905 — — — .89 .79 .31 .09
1906–15 — — .91 .89 .78 .23 .08
1916–25 — .87 .90 .90 .68 .22 —
1926–35 .55 .87 .90 .86 .64 — —
1936–45 .50 .87 .88 .86 — — —
1946–55 .47 .85 .87 — — — —
1956–65 .61 .83 — — — — —
1966–75 .60 — — — — — —
% Change 9.1 –4.6 –4.4 –3.4 –20.0 –55.1 –57.9
Females 1866–75 — — — — — — .03
1876–85 — — — — — .10 .04
1886–95 — — — — .23 .13 .04
1896–1905 — — — .32 .34 .13 .03
1906–15 — — .34 .45 .41 .11 .03
1916–25 — .31 .41 .51 .40 .12 —
1926–35 .33 .33 .48 .56 .44 — —
1936–45 .32 .43 .62 .68 — — —
1946–55 .38 .61 .73 — — — —
1956–65 .56 .69 — — — — —
1966–75 .59 — — — — — —
% Change 78.8 122.6 114.7 112.5 91.3 20.0 0

Average Weekly Hours Worked per Worker at Age (in Years)
Sex Year Born 15–24 25–34 35–44 45–54 55–64 65–74 75–84
Males 1866–75 — — — — — — 38.62
1876–85 — — — — — 42.07 34.13
1886–95 — — — — 43.95 37.35 32.56
1896–1905 — — — 44.96 42.53 34.88 29.59
1906–15 — — 45.22 43.92 42.02 33.06 28.64
1916–25 — 44.47 44.44 43.43 41.50 32.89 —
1926–35 40.49 43.84 44.02 43.06 41.69 — —
1936–45 33.94 43.19 43.58 43.93 — — —
1946–55 32.10 42.46 44.20 — — — —
1956–65 34.80 43.09 — — — — —
1966–75 33.46 — — — — — —
% Change –17.4 –3.1 –2.3 –2.3 –5.1 –21.8 –25.8
Females 1866–75 — — — — — — 36.34
1876–85 — — — — — 37.56 32.37
1886–95 — — — — 38.10 32.17 31.08
1896–1905 — — — 38.58 36.04 30.36 25.08
1906–15 — — 38.32 36.62 35.77 27.64 24.51
1916–25 — 38.14 35.79 36.00 34.73 27.94 —
1926–35 37.71 35.45 34.94 35.30 34.98 — —
1936–45 33.25 34.72 34.79 37.12 — — —
1946–55 31.48 35.47 36.91 — — — —
1956–65 31.57 37.14 — — — — —
1966–75 30.53 — — — — — —
% Change –19.0 –2.6 –3.7 –3.8 –8.2 –25.6 –32.6
Table 13
. . . And for the Hours Worked per Worker

Table 14
Possible Factors Behind Work Reallocations
In the United States, 1950–90
% of Population in Each Marital Status Category
Average
Index of Real Monthly Total Married With Spouse
Compensation* Social Security Fertility
Year (1982=100) Beneﬁt (1990 $) Rate** Present Absent Single Widowed Divorced
1950 49.4 238 3,337 64.45 4.01 21.10 8.25 2.24
1960 68.8 327 3,449 65.45 3.87 20.12 8.01 2.55
1970 91.3 397 2,480 61.33 3.86 23.23 8.17 3.41
1980 99.5 541 1,840 57.95 2.25 25.98 7.62 6.20
1990 103.8 603 2,081 53.56 4.29 26.42 7.37 8.35
% Change
1950–90 110.1 153.4 –37.6 –16.9 7.0 25.2 –10.7 272.8
*This is an index of hourly compensation in the business sector, deﬂated by the consumer price index for all
urban consumers.
**The fertility rate for any year is the number of births that 1,000 females would have in their lifetime if, at each
age, they experienced that year's birthrate.
Sources: See Appendix A.

A Decomposition of Average Weekly Hours
Worked per Person
Hours per Person Recalculated With
Actual Hours 1950 1950
Year per Person Weights Hours
1950 22.03 22.03 22.03
1960 20.97 21.40 21.57
1970 20.55 21.93 20.92
1980 22.00 23.30 20.99
1990 23.62 25.00 21.50
% Change
1950–90 7.2 13.5 –2.4
Source of basic data: See Appendix A.

Introduction to General Equilibrium II:
Firms
1

What is a ﬁrm?
• A technology:
y = F (k, l) = Akαl1−α
for α ∈ (0, 1).
• Operational deﬁnition.
• We are in a static world: we will assume k to be constant.
2

Properties of the Technology I
From lectures in growth we know that:
1. Constant returns to scale.
2. Inputs are essential.
3. Marginal productivities are positive and decreasing.
4. Inada Conditions.
3

Problem of the Firm I
• Wants to maximize proﬁts given r and w
(we are taking the consumption good as the numeraire!):
π = Akαl1−α − rk − wl
• We take ﬁrst order conditions:
αAkα−1l1−α = r (1)
(1 − α) Akαl−α = w (2)
• We want to solve for k and l.
4

Problem of the Firm II
• We begin dividing (1) by (2):
αAkα−1l1−α
(1 − α) Akαl−α
=
r
w
or
α
1 − α
l
k
=
r
w
or
k =
w
r
α
1 − α
l (3)
5

Problem of the Firm III
• but if we substitute (3) in (2):
(1 − α) Akαl−α = w
(1 − α) A
µ
w
r
α
1 − α
l
¶α
l−α = w
(1 − α) A
µ
w
r
α
1 − α
¶α
= w
l disappears!
• You can check that the same happens with k if we substitute (3) in
(1).
• What is wrong?
6

Problem of the Firm IV
• We have constant returns to scale.
• The size of the firm is indeterminate: we can have just one!
• To see that remember that profits are always zero if firm maximizes:
π = Akαl1−α − rk − wl
= Akαl1−α − αAkα−1l1−αk − (1 − α) Akαl−αl = 0
• So the firms really only pick the labor-capital ratio given relative prices:
l
k
=
α
1 − α
r
w
7

Problem of the Firm V
• In equilibrium, markets clear so:
l (r, w) = ls (w)
k (r, w) = kf
r = αAk (r, w)α−1 l (r, w)1−α
w = (1 − α) Ak (r, w)α
l (r, w)−α
• We have a system of four equations in four unknowns.
8

What are we missing?
• A lot.
• Wages are a lot of time diﬀerent from marginal productivites.
• Reasons for that:
1. Eﬃciency Wages: Shapiro-Stiglitz (1984).
2. Bargaining: Nash (1950).
3. Monopoly rents: Holmes and Schmitz (2001).
4. Sticky wages: Taylor (1980).
9

Screwmen
56
Longshoremen
50
40
16
Draymen
30
Coal
Wheelers
40
Carriage
20
Beer
21
19
34
33
39
29
Beer
33
29
Building
Trades
33
20
Rail
Dock
Dock
Rail
Freight Handlers
Teamsters Machinists Engineers
UNION
UNION
MFG. UNION MFG. UNION MFG.
Machine
Shops
Hourly Wage Rates (Cents) in Union Agreements and in Manufacturing Establishments, 1904–5
Trimming
Grain
General (range) Inside Outside
Source: Lee 1906
Dock Workers Were Paid More Than Most Other Workers in New Orleans . . .

Introduction to General Equilibrium III:
Government
1

What is the Government?
• Operational deﬁnition: takes taxes and spends them.
G = T
T = τlw
• No public debt.
• Why?
2

New Problem of the Household
• Problem for Robinson is now
max
c,l
u (c, 1 − l)
s.t. c =
³
1 − τl
´
wl + rk
• Why the new last term?
• FOC:
−
ul
uc
=
³
1 − τl
´
w
3

• u (c, l) = log c + γ log (1 − l)
• FOC+Budget constraint:
γ
c∗
1 − l∗
=
³
1 − τl
´
w
c∗ =
³
1 − τl
´
wl∗ + rk
• Then:
l∗ =
³
1 − τl
´
w − rk
(1 + γ)
³
1 − τl
´
w
4

• Taxes aﬀect labor supply!!!
• How important is the eﬀect?
• Two Examples:
1. Tax reform of 1986.
2. Why do Americans work so much more than Europeans?
5

Why Americans Work So Much
Edward C. Prescott
3
output was probably significantly larger than normal
and there may have been associated problems with the
market hours statistics. The earlier period was selected
because it is the earliest one for which sufficiently good
data are available to carry out the analysis. The relative
numbers after 2000 are pretty much the same as they
were in the pretechnology boom period 1993–96.
I emphasize that my labor supply measure is hours
worked per person aged 15–64 in the taxed market sec-
tor. The two principal margins of work effort are hours
actually worked by employees and the fraction of the
working-age population that works. Paid vacations, sick
leave, and holidays are hours of nonworking time. Time
spent working in the underground economy or in the
home sector is not counted. Other things equal, a country
with more weeks of vacation and more holidays will
have a lower labor supply in the sense that I am using
the term. I focus only on that part of working time for
which the resulting labor income is taxed.
Table 1 reports the G-7 countries’ output, labor sup-
ply, and productivity statistics relative to the United
States for 1993–96 and 1970–74. The important obser-
vation for the 1993–96 period is that labor supply (hours
per person) is much higher in Japan and the United States
than it is in Germany, France, and Italy. Canada and the
United Kingdom are in the intermediate range.Another
observation is that U.S. output per person is about 40
percent higher than in the European countries, with most
of the differences in output accounted for by differences
in hours worked per person and not by differences in
productivity, that is, in output per hour worked. Indeed,
the OECD statistics indicate that French productivity is
10 percent higher than U.S. productivity. In Japan, the
output per person difference is accounted for by lower
productivity and not by lower labor supply.
Table 1 shows a very different picture in the 1970–74
period. The difference is not in output per person. Then,
European output per person was about 70 percent of the
U.S. level, as it was in 1993–96 and is today. However,
the reason for the lower output in Europe is not fewer
market hours worked, as is the case in the 1993–96
period, but rather lower output per hour. In 1970–74,
Europeans worked more thanAmericans. The exception
is Italy. What caused these changes in labor supply?
Theory Used
To account for differences in the labor supply, I use the
standard theory used in quantitative studies of business
cycles (Cooley 1995), of depressions (Cole and Ohanian
1999 and Kehoe and Prescott 2002), of public finance
issues (Christiano and Eichenbaum 1992 and Baxter
and King 1993), and of the stock market (McGrattan
and Prescott 2000, 2003 and Boldrin, Christiano, and
Fisher 2001). In focusing on labor supply, I am follow-
ing Lucas and Rapping (1969), Lucas (1972), Kydland
and Prescott (1982), Hansen (1985), and Auerbach and
Kotlikoff (1987).
This theory has a stand-in household that faces a
labor-leisure decision and a consumption-savings de-
cision. The preferences of this stand-in household are
ordered by
(1)
Variable c denotes consumption, and h denotes hours of
labor supplied to the market sector per person per week.
Time is indexed by t. The discount factor 0 < <� 1
Table 1
Output, Labor Supply, and Productivity
In Selected Countries in 1993–96 and 1970–74
Relative to United States (U.S. = 100)
Output Hours Worked Output per
Period Country per Person* per Person* Hour Worked
1993–96 Germany 74 75 99
France 74 68 110
Italy 57 64 90
Canada 79 88 89
United Kingdom 67 88 76
Japan 78 104 74
United States 100 100 100
1970–74 Germany 75 105 72
France 77 105 74
Italy 53 82 65
Canada 86 94 91
United Kingdom 68 110 62
Japan 62 127 49
United States 100 100 100
*These data are for persons aged 15–64.
Sources: See Appendix.
E c ht
t t
t
� �(log log( )) .+ −





=
∞
∑ 100
0

Why Americans Work So Much
Edward C. Prescott
7
that the average labor supply (excluding the two outlier
observations) is close to the actual value for the other
12 observations.
Actual and Predicted Labor Supplies
Table 2 reports the actual and predicted labor supplies
for the G-7 countries in 1993–96 and 1970–74. For the
1993–96 period, the predicted values are surprisingly
close to the actual values with the average difference
being only 1.14 hours per week. I say that this number
is surprisingly small because this analysis abstracts from
labor market policies and demographics which have con-
sequences for aggregate labor supply and because there
are signiﬁcant errors in measuring the labor input.
The important observation is that the low labor sup-
plies in Germany, France, and Italy are due to high tax
rates. If someone in these countries works more and
produces 100 additional euros of output, that individual
gets to consume only 40 euros of additional consumption
and pays directly or indirectly 60 euros in taxes.
In the 1970–74 period, it is clear for Italy that some
factor other than taxes depressed labor supply. This
period was one of political instability in Italy, and quite
possibly cartelization policies reduced equilibrium labor
supply as in the Cole and Ohanian (2002) model of the
U.S. economy in the 1935–39 period. The overly high
prediction for labor supply for Japan in the 1970–74
period may in signiﬁcant part be the result of my util-
ity function having too little curvature with respect to
leisure, and as a result, the theory overpredicts when
the effective tax rate on labor income is low. Another
possible reason for the overprediction may be a measure-
ment error. The 1970–74 Japanese labor supply statistics
are based on establishment surveys only because at that
time household surveys were not conducted. In Japan
the household survey gives a much higher estimate of
hours worked in the period when both household- and
establishment-based estimates are available. In the other
Table 2
Actual and Predicted Labor Supply
In Selected Countries in 1993–96 and 1970–74
Labor Supply* Differences
Prediction Factors
(Predicted Consumption/
Period Country Actual Predicted Less Actual) Tax Rate � Output (c/y)
1993–96 Germany 19.3 19.5 .2 .59 .74
France 17.5 19.5 2.0 .59 .74
Italy 16.5 18.8 2.3 .64 .69
Canada 22.9 21.3 –1.6 .52 .77
United Kingdom 22.8 22.8 0 .44 .83
Japan 27.0 29.0 2.0 .37 .68
United States 25.9 24.6 –1.3 .40 .81
1970–74 Germany 24.6 24.6 0 .52 .66
France 24.4 25.4 1.0 .49 .66
Italy 19.2 28.3 9.1 .41 .66
Canada 22.2 25.6 3.4 .44 .72
United Kingdom 25.9 24.0 –1.9 .45 .77
Japan 29.8 35.8 6.0 .25 .60
United States 23.5 26.4 2.9 .40 .74
*Labor supply is measured in hours worked per person aged 15–64 per week.
Sources: See Appendix.

How Does the Government Behave?
• Economist use their tools to understand how governments behave.
• Political Economics (diﬀerent than Political Economy).
• Elements:
1. Rational Agents.
2. Optimization.
3. Equilibrium outcomes.
6

Overall Questions
• How do we explain differences and similarities in observed economic
policy over time?
1. Why do countries limit free trade?
2. Why do countries levy inefficient taxes?
• Can we predict the effects of changing political arrangements:
1. What would happen if we abandon the electoral college?
2. What would happen if we introduce proportional representation?
7

Some Basic Results
• Arrow’s Impossibility Theorem.
• Median Voter’s Theorem.
• Probabilistic Voting.
8

Political Economics in Macro
• How taxes are ﬁxed?
• Time-Consistency Problems.
• Redistribution.
9

General Equilibrium
1

Now we are going to put everything together
• We have a household that decides how much to work, l, and how
much to consume, c to maximize utility. It takes as given the wage,
w and the interest rate r.
• We have a ﬁrm that decides how much to produce, y and how much
capital, k, and labor, l to hire. It takes as given the wage, w and the
interest rate r.
• We have a government (maybe not) that raises taxes, T, and spends
G.
• We are in a static world: we will assume k to be constant.
2

Allocations, Feasible Allocations, Government Policy and Price Systems
• An allocation is a set of value for production, y, work, l, capital, k
and consumption c.
• A feasible allocation is an allocation that is possible:
c + G = y = Akαl1−α
• A government policy is a set of taxes τl and government spending G.
• A price system is a set of prices w and r.
3

A Competitive Equilibrium
A Competitive Equilibrium is an allocation {y, l, k, c}, a price system {w, r}
and a government policy
n
τl, T
o
such that:
1. Given the price system and the government policy, households choose
l and c to maximize their utility.
2. Given the price system and the government policy, ﬁrms maximize
proﬁts, i.e. they αAkα−1l1−α = r and (1 − α) Akαl−α = w.
3. Markets clear:
c + G = y = Akαl1−α
4

Existence of an Equilibrium
• Does it exist an equilibrium?
• Tough problem.
• Shown formally by Arrow (Nobel Prize Winner 1972) and Debreu (No-
bel Prize Winner 1983).
• Uniqueness?
5

What do we get out of the concept of an Equilibrium?
• Consistency: we are sure that everyone is doing things that are com-
patible. Economics is only social science that is fully aware of this big
issue.
• We talk about Competitive Equilibrium but there are other concepts
of equilibrium: Ramsey Equilibrium, Nash Equilibrium, etc...
• It is a prediction about the behavior of the model. Theory CAN and
SHOULD be tested against the data. Some theories are thrown away.
6

Application I: WWII and the Increase in G
• During WWII government spending to finance the war effort increased
to levels unseen previously in the US.
• What are the predictions of the model for this increase in spending?
• The assumption that government spending is a pure loss of output
arguably makes sense here. Pure spending/diversion of resources in
short run. Positive effects more long-run and harder to measure.
7

Abel/Bernanke, Macroeconomics, © 2001 Addison Wesley Longman, Inc. All rights reserved
Figure 1.06 U.S. Federal government spending and tax
collections, 1869-1999

• Preferences u(c, l) = log c + γ (1 − l).
• Cobb-Douglas technology.
• Production possibilities (goods market):
c = y − G = Akαl1−α − G
• To simplify g = G/y. So:
c = (1 − g)Akαl1−α.
• Household utility maximization:
ul(c, l)
uc(c, l)
=
γ
1/c
= w ⇒ γc = w
8

Choice under Uncertainty
1

• In the previous chapter, we studied intertemporal choices.
• However, there is a second important dimension of choice theory: un-
certainty.
• Life is also full of uncertainty: Will it rain tomorrow? Who will win
the next election? What will the stock market do next period?
• How do economist think about uncertainty?
• Von Neumann, Morgenstern, Debreu, Arrow, and Savage.
2

Simple Example I
• Flip a coin.
• Two events: s1 = {Heads} and s2 = {Tails} .
• Set of possible events S = {s1, s2} .
• Heads with probability π (s1) = p, tails with π (s2) = 1 − p.
• If heads, consumption is c (s1), if tails consumption is c (s2).
• Utility function is u(c (si)) for i = {1, 2} .
3

Simple Example II
• Under certain technical conditions, there is a Expected or Von Neumann-
Morgenstern utility function:
U(c (si)) = Eu(c (si)) = π (s1) u(c (s1)) + π (s2) u(c (s2))
• Linear in probabilities.
• Should you gamble? Role of the shape of the utility function.
• Risk-neutrality, risk-loving, risk-aversion.
4

More General Case
• We have n diﬀerent events:
Eu(c (si)) =
nX
i=1
π (si) u(c (si))
• Note that now consumption is a function mapping events into quan-
tities.
• π (si) can be objective or subjective.
5

Time and Uncertainty
• We can also add a time dimension.
• An event history st = (s0, s1, ..., st) .
• Then st ∈ St = S × S × ... × S
• Probabilities π
³
st
´
.
• Utility function:
∞X
t=0
X
st∈St
βtπ
³
st
´
u
³
c
³
st
´´
6

A Simpler Example: a Two Period World
• First period, only one event s0 with π
³
s0
´
= 1.
• Second period, n events, with probabilities π
³
s1
´
.
• Then, utility is:
u(c (s0)) + β
X
s1∈S1
π
³
s1
´
u
³
c
³
s1
´´
7

Markets
• Goods are also indexed by events.
• One good: endowment process e
³
st
´
.
• We introduce a set of one-period contingent securities b
³
st, st+1
´
for
all st ∈ St and st+1.
• Interpretation: b
³
st, st+1
´
pays one unit of good if and only if the
current history is st and tomorrow’s event is st+1.
• What are these securities in the real world?
8

Price of Securities
• Price of b
³
st, st+1
´
: q
³
st, st+1
´
.
• Quantity of b
³
st, st+1
´
: a
³
st, st+1
´
.
• Budget constraint:
c
³
st
´
+
X
st+1∈S
q
³
st, st+1
´
a
³
st, st+1
´
= e
³
st
´
+ a
³
st−1, st
´
•
n
q
³
st, st+1
´
a
³
st, st+1
´o
st+1∈S
is the portfolio of the household.
9

Equilibrium
A Sequential Markets equilibrium is an allocation
n
c∗
³
st
´
, a∗
³
st, st+1
´o∞
t=0,st∈St
and prices
n
q∗
³
st, st+1
´o∞
t=0,st∈St such that:
1. Given prices, the allocation solves:
max
∞X
t=0
X
st∈St
βtπ
³
st
´
u
³
c
³
st
´´
s.t. c
³
st
´
+
X
st+1∈S
q
³
st, st+1
´
a
³
st, st+1
´
= e
³
st
´
+ a
³
st−1, st
´
2. Markets clear c
³
st
´
= e
³
st
´
.
10

Characterization of the Equilibrium
1. We can prove existence of a Sequential Markets equilibrium.
2. There are other, equivalent, market structures.
3. The two fundamental welfare theorems hold.
4. Note, however, how we need markets for all goods under all possible
events!
11

Is Our Representation of Uncertainty a Good One?
• Uncertainty aversion: Ellsberg’s paradox.
• Diﬀerent models of the world.
• Robustness of our decisions.
• Unawareness.
12

Putting Theory to Work: Asset Pricing
• We revisit our two periods example.
• Preferences:
u(c (s0)) + β
X
s1∈S1
π
³
s1
´
u
³
c
³
s1
´´
• Budget constraints:
c (s0) +
X
s1∈S
q
³
s0, s1
´
a
³
s0, s1
´
= e
³
s0
´
c (s1) = e (s1) + a
³
s0, s1
´
for all s1 ∈ S
13

Problem of the Household
• We write the Lagrangian:
u(c (s0)) + β
X
s1∈S1
π
³
s1
´
u
³
c
³
s1
´´
+λ (s0)

e
³
s0
´
− c (s0) +
X
s1∈S
q
³
s0, s1
´
a
³
s0, s1
´


+
X
s1∈S1
λ
³
s1
´ ³
e (s1) + a
³
s0, s1
´
− c (s1)
´
• We take ﬁrst order conditions with respect to c (s0) , c (s1), and
a
³
s0, s1
´
.
14

Solving the Problem
• FOCs
u0(c (s0)) = λ (s0)
βπ
³
s1
´
u0(c
³
s1
´
) = λ
³
s1
´
for all s1 ∈ S
λ (s0) q
³
s0, s1
´
= λ
³
s1
´
for all s1 ∈ S
• Then:
q
³
s0, s1
´
= π
³
s1
´
β
u0
³
c
³
s1
´´
u0 (c (s0))
• Fundamental equation of Asset Pricing.
15

The Stochastic Discount Factor
• Stochastic discount factor (or pricing kernel):
m
³
s1
´
= β
u0
³
c
³
s1
´´
u0 (c (s0))
• Note that:
Em
³
s1
´
=
X
s1∈S1
π
³
s1
´
m
³
s1
´
= β
X
s1∈S1
π
³
s1
´ u0
³
c
³
s1
´´
u0 (c (s0))
16

Pricing Redundant Securities
• With our framework we can price any security.
• For example, an uncontingent bond:
q
³
s0
´
=
X
s1∈S1
q
³
s0, s1
´
= β
X
s1∈S1
π
³
s1
´ u0
³
c
³
s1
´´
u0 (c (s0))
• Generalize to very general ﬁnancial contracts:
p
³
s0, s1
´
= βπ
³
s1
´
x
³
s1
´ u0
³
c
³
s1
´´
u0 (c (s0))
17

Risk-Free Rate
• Note that:
q
³
s0
´
= Em
³
s1
´
• Then, the risk-free rate:
Rf
³
s1
´
=
1
q
³
s0
´ =
1
Em
³
s1
´
or ERf
³
s1
´
m
³
s1
´
= 1.
18

Example of Financial Contracts
1. Stock: buy at price p (s0) , delivers a dividend d
³
s1
´
, sells at p (s1)
p
³
s0
´
= β
X
s1∈S1
π
³
s1
´ ³
p (s1) + d
³
s1
´´ u0
³
c
³
s1
´´
u0 (c (s0))
2. Call option: buy at price o (s0) the right to buy an asset at price K1.
Price of asset J
³
s1
´
o
³
s0
´
= β
X
s1∈S1
π
³
s1
´
max


³
J
³
s1
´
− K1
´ u0
³
c
³
s1
´´
u0 (c (s0))
, 0


19

Non Arbitrage
• A lot of ﬁnancial contracts are equivalent.
• From previous results, we derive a powerful idea: absence of arbitrage.
• Empirical evidence regarding non arbitrage.
• Possible limitations to non arbitrage conditions.
• Related idea: spanning of non-traded assets.
20

Simple Example
• u(c) = log c, β = 0.99
• e
³
s0
´
= 1, e (s1 = high) = 1.1, e (s1 = low) = 0.9.
• π (s1 = high) = 0.5, π (s2 = low) = 0.5.
21

• Equilibrium prices:
q
³
s0, s1 = high
´
= 0.99 ∗ 0.5 ∗
1
1.1
1
1
= 0.45
q
³
s0, s1 = low
´
= 0.99 ∗ 0.5 ∗
1
0.9
1
1
= 0.55
q
³
s0
´
= 0.45 + 0.55 = 1
• Note how the price is diﬀerent from a naive adjustment by expectation
and discounting:
qnaive
³
s0, s1 = high
´
= 0.99 ∗ 0.5 ∗ 1 = 0.495
qnaive
³
s0, s1 = low
´
= 0.99 ∗ 0.5 ∗ 1 = 0.495
qnaive
³
s0
´
= 0.495 + 0.495 = 0.99
22

• Why is q
³
s0, s1 = high
´
< q
³
s0, s1 = low
´
?
• Two forces:
1. Discounting β.
2. Ratio of marginal utilities:
u0(c(s1))
u0(c(s0))
.
• Covariance is key.
23

Risk Correction I
We recall three facts:
1. q
³
s0
´
= Em
³
s1
´
.
2. p
³
s0, s1
´
= Em
³
s1
´
x
³
s1
´
.
3. cov(xy) = E(xy) − E(x)E(y).
24

Risk Correction II
Then:
p
³
s0, s1
´
= Em
³
s1
´
Ex
³
s1
´
+ cov
³
m
³
s1
´
x
³
s1
´´
or
p
³
s0, s1
´
=
Ex
³
s1
´
Rf
³
s1
´ + cov
³
m
³
s1
´
x
³
s1
´´
=
Ex
³
s1
´
Rf
³
s1
´ + cov

β
u0
³
c
³
s1
´´
u0 (c (s0))
x
³
s1
´


=
Ex
³
s1
´
Rf
³
s1
´ + β
cov
³
u0
³
c
³
s1
´´
x
³
s1
´´
u0 (c (s0))
25

Risk Correction III
Now we can see how if:
1. If cov
³
m
³
s1
´
x
³
s1
´´
= 0 ⇒ p
³
s0, s1
´
=
Ex(s1)
Rf(s1)
, not adjustment for
risk.
2. If cov
³
m
³
s1
´
x
³
s1
´´
> 0 ⇒ p
³
s0, s1
´
>
Ex(s1)
Rf(s1)
, premium for risk
(insurance).
3. If cov
³
m
³
s1
´
x
³
s1
´´
< 0 ⇒ p
³
s0, s1
´
<
Ex(s1)
Rf(s1)
, discount for risk
(speculation).
26

Utility Function and the Risk Premium
• We see how risk depends of marginal utilities:
1. Risk-neutrality: if utility function is linear, you do not care about
var (c) .
2. Risk-loving: if utility function is convex you want to increase var (c).
3. Risk-averse: if utility function is concave you want to reduce var (c).
• It is plausible that household are (basically) risk-averse.
27

CRRA Utility Functions I
• Market price of risk has been roughly constant over the last two cen-
turies.
• This observation suggests that risk aversion should be relatively con-
stant over the wealth levels.
• This is delivered by constant relative risk aversion utility function:
c1−σ
1 − σ
• Note that when σ = 1, the function is log ct (you need to take limits
and apply L’Hˆopital’s rule).
28

CRRA Utility Functions II
• σ plays a dual role controlling risk-aversion and intertemporal substi-
tution.
• Coeﬃcient of Relative Risk-aversion:
−
u00(c)
u0(c)
c = σ
• Elasticity of Intertemporal Substitution:
−
u(c2)/u(c1)
c2/c1
d (c2/c1)
d (u(c2)/u(c1))
=
1
σ
• Advantages and disadvantages.
29

Size of σ
• Most evidence suggests that σ is low, between 1 and 3. At most 10.
• Types of evidence:
1. Questionnaires.
2. Experiments.
3. Econometric estimates from observed behavior.
• A powerful arguments from international comparisons.
30

A Small Detour
• Note that all we have said can be applied to the trivial case without
uncertainty.
• In that situation, there is only one security, a bond, with price:
q1 = β
u0(c1)
u0(c0)
• And the interest rate is:
R1 =
1
q1
=
1
β
u0(c0)
u0(c1)
31

Pricing Securities in the Solow Model
• Assume that the utility is CRRA and that we are in a BGP with γ = g.
• Then:
R =
1
β
Ã
c
(1 + g) c
!−σ
=
(1 + g)σ
β
• Or in logs: r ' 1 + σg − β, i.e., the real interest rate depends on the
rate of growth of technology, the readiness of households to substitute
intertemporally, and on the discount factor.
• Also, σ must be low to reconcile small international diﬀerences in the
interest rate and big diﬀerences in g.
32

Mean-Variance Frontier
• The pricing condition for a contract i with price 1 and yield Ri
³
s1
´
is:
1 = Em
³
s1
´
Ri
³
s1
´
• Then:
1 = Em
³
s1
´
ERi
³
s1
´
+ cov
³
m
³
s1
´
Ri
³
s1
´´
or:
1 = Em
³
s1
´
ERi
³
s1
´
+
cov
³
m
³
s1
´
Ri
³
s1
´´
sd
³
m
³
s1
´´
sd
³
Ri
³
s1
´´sd
³
m
³
s1
´´
sd
³
Ri
³
s1
´´
33

• The coeﬃcient of correlation between to random variables is:
ρm,Ri
=
cov
³
m
³
s1
´
Ri
³
s1
´´
sd
³
m
³
s1
´´
sd
³
Ri
³
s1
´´
• Then, we have:
1 = Em
³
s1
´
ERi
³
s1
´
+ ρm,Ri
sd
³
m
³
s1
´´
sd
³
Ri
³
s1
´´
• Or:
ERi
³
s1
´
= Rf − ρm,Ri
sd
³
m
³
s1
´´
Em
³
s1
´ sd
³
Ri
³
s1
´´
34

• Since ρm,Ri
∈ [−1, 1] :
¯
¯
¯ERi
³
s1
´
− Rf
¯
¯
¯ ≤
sd
³
m
³
s1
´´
Em
³
s1
´ sd
³
Ri
³
s1
´´
• This relation is known as the Mean-Variance frontier.
• Relation between mean and variance of an asset: “How much return
can you get for a given level of variance?”
•
sd(m(s1))
Em(s1)
can be interpreted as the market price of risk.
• Any investor would hold assets within the mean-variance region.
35

The Sharpe Ratio
• Another way to represent the Mean-Variance frontier is:
¯
¯
¯
¯
¯
¯
ERi
³
s1
´
− Rf
sd
³
Ri
³
s1
´´
¯
¯
¯
¯
¯
¯
≤
sd
³
m
³
s1
´´
Em
³
s1
´
• This relation is known as the Sharpe Ratio.
• It answers the question: “How much more mean return can I get by
shouldering a bit more volatility in my portfolio?”
36

The Equity Premium Puzzle I
• Assume a CRRA utility function.
• Then, m
³
s1
´
=
µ
c(s1)
c(s0)
¶−σ
• A good approximation of
sd(m(s1))
Em(s1)
is (forget about the algebra de-
tails):
σsd
³
∆ ln c
³
st
´´
37

The Equity Premium Puzzle II
• Let us go to the data and think about the stock market (i.e. Ri
³
s1
´
is the yield of an index) versus the risk free asset (the U.S. treasury
bill).
• Average return from equities in XXth century: 6.7%. From bills 0.9%.
• Standard deviation of equities: 16%.
• Standard deviation of ∆ ln c
³
st
´
: 1%.
38

The Equity Premium Puzzle III
• Then:
¯
¯
¯
¯
¯
6.7% − 0.9%
16%
¯
¯
¯
¯
¯
= 0.36 ≤ σ1%
that implies a σ of at least 36!
• But we argued before that σ is at most 10.
• This observation is known as the Equity Premium Puzzle (Mehra and
Prescott, 1985)
39

Answers to Equity Premium Puzzle
1. Returns from the market have been odd. For example, if return from
bills had been around 4% and returns from equity 5%, you would only
need a σ of 6.25. Some evidence related with the impact of inﬂation.
2. There were important distortions on the market. For example regula-
tions and taxes.
3. People is an order of magnitude more risk averse that we think. Epstein-
Zin preferences.
4. The model is deeply wrong.
40

Random Walks I
• Can we predict the market?
• Remember that the price of a share was
p
³
s0
´
= β
X
s1∈S1
π
³
s1
´ ³
p (s1) + d
³
s1
´´ u0
³
c
³
s1
´´
u0 (c (s0))
or:
p
³
s0
´
= βE
³
p (s1) + d
³
s1
´´ u0
³
c
³
s1
´´
u0 (c (s0))
41

Random Walks II
• Now, suppose that we are thinking about a short period of time, i.e.
β ≈ 1 and that ﬁrms do not distribute dividends (not a bad approxi-
mation because of tax reasons):
p
³
s0
´
= Ep (s1)
u0(
³
c
³
s1
´´
u0 (c (s0))
• If in addition
u0(c(s1))
u0(c(s0))
does not change (either because utility is linear
or because of low volatility of consumption):
p
³
s0
´
= Ep (s1) = p
³
s0
´
+ ε0
42

Random Walks III
• p
³
s0
´
= p
³
s0
´
+ ε0 is called a Random Walk.
• The best forecast of the price of a share tomorrow is today’s price.
• Can we forecast future movements of the market? No!
• We can generalize the idea to other assets.
43

Main Ideas of Asset Pricing
1. Non-arbitrage.
2. Risk-free rate is r ' 1 + σg − β.
3. Risk is not important by itself: the key is covariance.
4. Mean-Variance frontier.
5. Equity Premium Puzzle.
6. Random walk of asset prices.
44

• Firm profit maximization:
(1 − α) Akαl−α = w
• Equate and impose goods market clearing:
γc = (1 − α) Akαl−α
⇒ γ(1 − g)Akαl1−α = (1 − α) Akαl−α
⇒ l =
1 − α
γ(1 − g)
• Government spending has a pure income effect here (since financed
by lump sum taxes). Increases labor supply.
9

• Solve for rest of allocation:
y = Akα
"
1 − α
γ(1 − g)
#1−α
c = (1 − g)y = (1 − g)αkα
"
1 − α
γ
#1−α
• Output increases with g, consumption decreases.
10

• Solve for wages and interest rates:
w = (1 − α) Akα
"
1 − α
γ(1 − g)
#−α
r = αAkα−1
"
1 − α
γ(1 − g)
#1−α
• Wages decrease with g, interest rates increase.
11

Summing Up
• Following increase in g = G/Y , the model predicts an increase in
(y, l, r), decrease in (c, w).
• Private consumption spending is “crowded out” by increased govern-
ment spending.
• Output increases but loss of welfare as both c, 1 − l fall.
• These predictions match US experience of WWII.
12

Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 5-8
Figure 5.7 GDP, Consumption, and
Government Expenditures

Figure 15.04 Deficits and primary deficits: Federal, state, and
local, 1940-1998

Figure 1.02 Average labor productivity in the United States,
1900-1998

What Does This Analysis Miss?
• Government debt. A large fraction of the wartime spending was fi-
nanced by government debt.
Deficit/GDP ratio hit 24% by 1944.
• Debt allows for intertemporal substitution of resources and smooth-
ing burden of taxation. If needed to increase (distortionary) taxes to
finance full war spending, production would have been less.
• Increased productivity. Wartime mobilization of production increased
labor productivity dramatically.
• Led to larger increase in production than our model suggests.
13

Application II: Skill Biased Technical Change and Inequality
• Large literature documenting increase in income and wealth inequality
in the US.
• Started in the 1970s and continues today.
• At same time, has been a large increase in the returns to education:
1. Average wages of college graduates from increased by 60% for
males and 90% for females from 1963 to 2002.
2. Average wages of high school graduates only increased by 20% for
males and 50% for females over same period.
14

Main explanation:
• Skill-biased technical change.
• Skilled and unskilled labor are eﬀectively diﬀerent labor markets.
• Productivity changes have increased the relative demand for skilled
labor.
15

Analysis of Skill-Biased Change
• Extend the previous to two types of households: skilled and unskilled.
• Households: Assume both skilled and unskilled workers have same
preferences:
u(c, l) = c −
l2
2
• Assume skilled workers own a share β of the capital stock, unskilled a
share (1 − β).
• Wages ws for skilled wu for unskilled.
16

Skilled Household Problem
max
ls
(
lsws + βrK −
ls
2
2
)
Optimality conditions:
−
ul
uc
= ls = ws
• So ls = ws, cs = w2
s + βrK.
• Unskilled household problem is equivalent:
lu = wu, cu = w2
u + (1 − β) rK.
17

Firms
• Assume representative ﬁrm hires both skilled and unskilled labor. Each
has diﬀerent productivity (zs, zu).
• Firm substitutes between skilled and unskilled for total labor input.
l = (zslρ
s + zulρ
u)1/ρ
where 0 < ρ < 1.
• Thus production is:
y = kαl1−α
18

• Firms maximize proﬁts:
kαl(ls, lu)1−α − rk − wsls − wulu
• FOC’s — each type paid its marginal product:
(1 − α)kαl−α (zslρ
s + zulρ
u)1/ρ−1
zslρ−1
s = ws
(1 − α)kαl−α (zslρ
s + zulρ
u)1/ρ−1
zulρ−1
u = wu
19

Characterize Equilibrium
• Divide ﬁrm FOC’s:
ws
wu
=
zsl
ρ−1
s
zul
ρ−1
u
⇒
log
ws
wu
= log
zs
zu
+ (ρ − 1) log
ls
lu
• ws
wu
, the skilled premium depends on:
1. Relative productivities: zs
zu.
2. Relative supplies: ls
lu
20

• Then:
γ ws
wu
= γ zs
zu
+ (ρ − 1) γ ls
lu
• Changes in skill-premium depend on:
1. Relative changes in productivities.
2. Relative changes in abundance of factors.
21

Why Did it Happen?
• Nature of modern science.
• Size of the Market.
• Economics of Superstars.
22

What Does This Analysis Miss?
• Captures broad aggregate facts. Misses on some dimensions.
• Consumption Inequality. Evidence that inequality in consumption was
less than inequality in income (Krueger and Perri, 2003). Here we
have greater consumption inequality (since β ≈ 1) :
cs
cu
=
w2
s + βrK
w2
u + (1 − β)rK
≈
µ
ws
wu
¶2
+
rK
w2
u
• Changes by Gender. Most dramatic eﬀects have been increase in fe-
male labor supply, especially in skilled labor. Hard to argue this was all
from skill-biased technical change. Composition eﬀects may be more
important. (Eckstein and Nagypal, 2004)
23

Welfare Theorems
1

Pareto Optimality
• An allocation is Pareto Optimal if there is no way to rearrange produc-
tion or reallocate goods so that someone is made better oﬀ without
making someone else worse oﬀ.
• Pareto Optimality 6= perfect state of the world or any concept like
that.
2

The Social Planner
• Let us imagine we have a powerful dictator, the Social Planner, that
can decide how much the households consume and work and how much
the ﬁrms produce.
• The Social Planner does not follow prices. But it understands oppor-
tunity cost.
• The Social Planner is benevolent. It searches for the best possible
allocation.
3

Social Planner’s Problem I
• Maximizes utility household given a level of government purchases G∗
max
c,l
u (c, 1 − l)
such that
c + G = Akαl1−α
G = G∗
k = k∗
• Note: we do not have prices in the budget constraint!!!
• Standard Maximization problem.
4

Social Planner’s Problem II
• We can rewrite the problem as:
max
l
u
³
Ak∗αl1−α − G∗, 1 − l
´
• First Order Condition with respect to l:
u0
c
³
Ak∗αl1−α − G∗, 1 − l
´
(1 − α) Ak∗αl−α
−u0
1−l
³
Ak∗αl1−α − G∗, 1 − l
´
= 0
5

Social Planner’s Problem III
• We rearrange as:
u0
1−l
³
Ak∗αl1−α − G∗, 1 − l
´
u0
c
³
Ak∗αl1−α − G∗, 1 − l
´ = (1 − α) Ak∗αl−α
• The lhs is the Marginal Rate of Substitution, MRS while the rhs is the
Marginal Rate of Transformation, MRT.
• Thus, optimality implies:
MRS = MRT
• Let’s look at it graphically.
6

The Big Question
• What is the relation between the solution to the Planners Problem
and the Competitive Equilibrium?
• Or equivalently, is the Competitive Equilibrium Pareto-Optimal?
• Why do we care about this question?
1. Positive reasons
2. Normative reasons.
7

The Intuition
• First think about the case when G∗ = τl = 0
• Look again at the Social Planner’s optimality condition
u0
1−l
³
Ak∗αl1−α, 1 − l
´
u0
c
³
´ = (1 − α) Ak∗αl−α
• Remember that the Household ﬁrst order condition was:
u0
1−l
³
Ak∗αl1−α − G∗, 1 − l
´
u0
c
³
Ak∗αl1−α − G∗, 1 − l
´ = w
8

• And that ﬁrms proﬁt maximization implied:
w = (1 − α) Ak∗αl−α
• First order conditions are equivalent!!!
9

The Formal Statement
• First Fundamental Welfare Theorem: under certain conditions, the
Competitive Equilibrium is Pareto Optimal.
• We have the converse.
• Second Fundamental Welfare Theorem: under certain conditions, a
Pareto optimum is a Competitive Equilibrium.
10

Some consequences
• First Fundamental Welfare Theorem states that, under certain condi-
tions, an allocation achieved by a market economy is Pareto-Optimal.
• Formalization of Adam Smith’s “invisible hand” idea.
• Strong theoretical point in favour of decentralized allocation mech-
anisms: prices direct agents to do what is needed to get a Pareto
optimum.
• Second Fundamental Welfare Theorem states what is the best way to
change allocations: redistribute income. Do not mess with prices!!!
11

How robust is the First Welfare theorem?
• Not too much.
• Plenty of reasons that deviate the allocation from a Pareto optimum:
1. Taxes.
2. Externalities.
3. Asymmetric Information.
4. Market Incompleteness.
5. Bounded Rationality of Agents.
12

What if taxes are not zero?
• Now think about the case when G∗ 6= 0, τl 6= 0
• Look again at the Social Planner’s optimality condition
u0
1−l
³
´
u0
c
³
´ = (1 − α) Ak∗αl−α
• But now the Household ﬁrst order condition is:
u0
1−l
³
Ak∗αl1−α − G∗, 1 − l
´
u0
c
³
Ak∗αl1−α − G∗, 1 − l
´ =
³
1 − τl
´
w
13

• And since that ﬁrms proﬁt maximization implied:
w = (1 − α) Ak∗αl−α
• First order conditions are NOT equivalent!!!
14

Externalities
• What is an externality? When an agents consumption or production
decision changes the production or consumption possibilities of other
agents.
• Externalities can be good or bad.
• Example:
1. Cities
2. Environment
15

Asymmetric Information
• Information is dispersed in society.
• We may want to change our behavior based on the information we
have.
• Akerlof-Spence-Stiglitz, Nobel Prize Winners 2001.
• Townsend and Prescott (1985).
16

Market Incompleteness
• We have assumed that we have complete markets.
• Every good can be traded.
• Is that a good representation of the world?
• Closely related with the problem of asymmetric information.
17

Bounded Rationality
• We have assumed that agents are “rational”.
• Is this a good hypothesis?
• In some sense, yes:
1. it is very powerful and more general that sometimes claimed.
2. it is simple.
• Are we rational? Can we process information accurately?
18

A rationality quiz
• Assume:
1. 1 in 100 people in the world are rational.
2. We have a test for rationality.
3. If someone is rational, it has a 99% chance of passing the test. If
someone is irrational, it has a 99% chance of failing.
4. My brother just passed the test.
5. My brother was selected randomly from the population.
• What is the probability that my brother is rational?
19

Answer
Pr(rational| pass) =
Pr(pass| rational) Pr(rational)
Pr(pass)
=
0.99 ∗ 0.01
0.99 ∗ 0.01 + 0.01 ∗ 0.99
=
1
2
Ask yourself (honestly): which answer I thought it was right?
20

Bounded rationality models
• Basic insight from Evolution Theory: we are not perfect machines, but
frozen DNA accidents.
• Intersection between Evolutionary Psychology and Economics.
• Models with agents have problems to compute and they do not really
know what they want.
• Problem: There is ONE way to be rational. There are MANY ways to
be irrational.
• Which one is a better modelling choice?
21

What happens when we deviate from the assumptions of the theorem?
• It is hard to say.
• Second-Best Theorem (Tony Lancaster).
• Basic implications for reforms.
• How do we think about the real world?
22

Putting our theory to work
• How do we allocate resources in society?
• Why is this important? a little bit of history
• Could Central Planning work? Mises, Hayek in the 20’s: NO
• Experience is rather clear that it did not, but maybe they just did not
apply the recipe properly.
• That was the idea behind Lange-Lerner proposals for a Market-based
socialism. Modern defenders of the idea: Roemer.
23

The intuition behind the idea
• What really matters is the use of prices, not private ownership.
• If somehow we can replicate the behavior of prices we’d be home free.
• Never really tried, but we have strong theoretical predictions against
it.
24

The problem of information
“The problem of rational economic order is determined precisely by the
fact that the knowledge of the circumstances of which we must make use
never exist in concentrated or integrated form, but solely as the dispersed
bits of incomplete knowledge which all the separate individuals possess...
The problem is thus in no way solved if we can show that all the facts, if
they were known to a single mind (as we hypothetically assume them to be
given to the observing economist) would uniquely determine the solution;
instead we must show how a solution is produced by the interactions of
people, each of whom possesses only partial knowledge”.
Friedrich von Hayek
25

The market as a way to process information
• At the end of the day the Welfare Theorems do not capture all the
advantages of market economies.
• They do not talk for instance about experimentation.
• Market economies are robust: they allow experimentation.
• Intuition from Biology.
• Example: Minitel in France versus Internet in the U.S.
26

Intertemporal Choice
1

• So far we have only studied static choices.
• Life is full of intertemporal choices: Should I study for my test today
or tomorrow? Should I save or should I consume now? Should I marry
this year or the next one?
• We will present a simple model: the Life-Cycle/Permanent Income
Model of Consumption.
• Developed by Modigliani (Nobel winner 1985) and Friedman (Nobel
winner 1976).
2

The Model
• Household, lives 2 periods.
• Utility function
u(c1, c2) = U(c1) + βU(c2)
where c1 is consumption in ﬁrst period of his life,c2 is consumption in
second period of his life, and β is between zero and one and measures
household’s degree of impatience.
• Income y1 > 0 in the ﬁrst period of life and y2 ≥ 0 in the second
period of his life.
3

Budget Constraint I
• Household can save some of his income in the ﬁrst period, or he can
borrow against his future income y2.
• Interest rate on both savings and on loans is equal to r. Let s denote
saving.
• Budget constraint in ﬁrst period of life:
c1 + s = y1
• Budget constraint in second period of his life:
c2 = y2 + (1 + r)s
4

Budget Constraint II
• Summing both budget constraints:
c1 +
c2
1 + r
= y1 +
y2
1 + r
= I
• We have normalized the price of the consumption good in the ﬁrst
period to 1. Price of the consumption good in period 2 is 1
1+r, which
is also the relative price of consumption in period 2, relative to con-
sumption in period 1.
• Gross interest rate 1 + r is the relative price of consumption goods
today to consumption goods tomorrow.
5

max
c1,c2
U(c1) + βU(c2)
s.t. c1 +
c2
1 + r
= I
• FOC:
U0(c1) = λ
βU0(c2) = λ
1
1 + r
• Then we get Euler Equation: U0(c1) = β (1 + r) U0(c2).
6

• If U(c) = log c, Euler Equation:
1
c1
= β (1 + r)
1
c2
⇒ c2 = β (1 + r) c1
• Note that:
c1 = I −
c2
1 + r
= I − βc1
and then c1 = 1
1+βI and c2 =
β(1+r)
1+β I.
• Then s = y1 − c1 = β
1+βy1 − 1
1+β
³
y2
1+r
´
7

Key Results
• Optimal consumption choice today: eat a fraction 1
1+β of total lifetime
income I today and save the rest for the second period of your life.
• What variables does current consumption depend on? y1, y2, r.
8

Comparative Statics: Income Changes
• What happens to consumption if y1 or y2 increases?
• Both c1 and c2 increase.
• Marginal propensity to consume out of current income or wealth
dc1
dy1
=
1
1 + β
> 0
• Marginal propensity to consume out of tomorrows income
dc1
dy2
=
1
(1 + β)(1 + r)
> 0
9

Comparative Statics: Changes in the Interest Rate
• Income effect: if a saver, then higher interest rate increases income
for given amount of saving. Increases consumption in first and second
period. If borrower, then income effect negative for c1 and c2.
• Substitution effect: gross interest rate 1 + r is relative price of con-
sumption in period 1 to consumption in period 2. c1 becomes more
expensive relative to c2. This increases c2 and reduces c1.
• Hence: for a saver an increase in r increases c2 and may increase
or decrease c1. For a borrower an increase in r reduces r1 and may
increase or decrease c2.
10

Borrowing Constraints
• So far: household could borrow freely at interest rate r.
• Now: assume borrowing constraints s ≥ 0.
• If household is a saver, nothing changes.
• If household would be a borrower without the constraint, then c1 = y1,
c2 = y2. He would like to have bigger c1, but he can’t bring any of his
second period income forward by taking out a loan. In this situation
ﬁrst period consumption does not depend on second period income or
the interest rate.
11

Extension of the Basic Model: Life Cycle Hypothesis
• We can extend to T periods: Franco Modigliani’ life-cycle hypothesis
of consumption
• Individuals want smooth consumption profile over their life. Labor
income varies substantially over lifetime, starting out low, increasing
until the 50’th year of a person’s life and then declining until 65, with
no labor income after 65.
• Life-cycle hypothesis: by saving (and borrowing) individuals turn a
very nonsmooth labor income profile into a very smooth consumption
profile.
12

• Main predictions:
1. current consumption depends on total lifetime income.
2. Saving should follow a very pronounced life-cycle pattern with bor-
rowing in the early periods of an economic life, signiﬁcant saving
in the high earning years from 35-50 and dissaving in retirement
years.
• Do we observe these predictions in the data?
13

Empirical puzzles
• Hump in consumption. Role of demographics and uncertainty.
• Excess sensitivity of consumption to income.
• Older household do not dissave to the extent predicted by the theory.
Several explanations:
1. Individuals are altruistic and want to leave bequests to their chil-
dren.
2. Uncertainty with respect to length of life and health status.
14

20 30 40 50 60 70 80 90
2000
2500
3000
3500
4000
4500
5000
5500
Figure 4.1: Total Expenditure
Age
1982-84$
20 30 40 50 60 70 80 90
800
1000
1200
1400
1600
1800
2000
Figure 4.2: Expenditures non Durables
Age
1982-84$
20 30 40 50 60 70 80 90
400
600
800
1000
1200
1400
1600
1800
Figure 4.3: Expenditures Durables
Age
1982-84$

20 30 40 50 60 70 80 90
1800
2000
2200
2400
2600
2800
3000
3200
3400
Figure 4.4: Total Expenditure, Adult Equivalent
Age
1982-84$
20 30 40 50 60 70 80 90
800
850
900
950
1000
1050
1100
1150
1200
1250
1300
Figure 4.5: Expenditures non Durables, Adult Equivalent
Age
1982-84$
20 30 40 50 60 70 80 90
400
500
600
700
800
900
1000
1100
Figure 4.6: Expenditures Durables, Adult Equivalent
Age
1982-84$

20 30 40 50 60 70 80 90
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
Figure 4.7: Total Expenditure, Adult Equivalent, by Education Groups
Age
High Education
Low Education
Benchmark

Application of the Theory I: Social Security in the Life-cycle model
• Social Security is an important ongoing debate.
• Two classes of questions:
1. Positive questions: What are the eﬀects of social security and its
possible reforms? What is the forecasted evolution of the current
system?
2. Normative questions: How should we organize social security?
• How does social security work in the U.S.?
15

• We want to distinguish:
1. Between sustainability of the current system and the optimal re-
forms.
2. Between pay-as-you-go versus fully funded system and private ver-
sus public systems.
3. Role of system as saving, insurance, and redistribution.
• Use simple life-cycle model to analyze some of these issues.
16

Social Security in the Life-cycle model
• Assume y2 = 0.
• Without social security (or a fully funded system).
c1 =
y1
1 + β
c2 =
β(1 + r)y1
1 + β
s =
βy1
1 + β
17

Pay As-You-Go Social Security System
• Introduce a pay as-you-go social security system: currently working
generation pays payroll taxes, whose proceeds are used to pay the
pensions of the currently retired generation
• Payroll taxes at rate τ in ﬁrst period. After tax wage is (1 − τ)y1.
Currently in US τ = 12.4%
• Social security payments SS in second period: assume that population
grows at rate n and pre-tax-income grows at rate g.
• Social security system balances its budget:
SS = (1 + g)(1 + n)τy1
18

• Household’s budget constraints:
c1 + s = (1 − τ)y1
c2 = (1 + r)s + SS
• Intertemporal budget constraint:
c1 +
c2
1 + r
= (1 − τ)y1 +
SS
1 + r
= I
• Maximizing utility subject to the budget constraint yields:
c1 =
I
1 + β
c2 =
β
1 + β
(1 + r)I
19

• Since SS = (1 + g)(1 + n)τy1:
I = (1 − τ)y1 +
SS
1 + r
= (1 − τ)y1 +
(1 + g)(1 + n)τy1
1 + r
= y1 −
Ã
1 −
(1 + g)(1 + n)
1 + r
!
τy1
= ˜y1
• Hence:
c1 =
˜y1
1 + β
c2 =
β
1 + β
(1 + r)˜y1
20

• Consumption in both periods is higher with social security than without
if and only if ˜y1 > y1, i.e. if and only if
(1+g)(1+n)
1+r > 1. People are
better oﬀ with social security if
(1 + g)(1 + n) > 1 + r
• Intuition: If people save by themselves for retirement, return on their
savings equals 1 + r. If they save via a social security system, return
equals (1 + n)(1 + g).
21

Numbers
• n = 1%, g = 2%.
• What is a good estimate r (in real terms)?
• Historical Record:
1900-2000 1970-2000
Equities 6.7% 7.2%
Bonds 1.6% 4.1%
Bills 0.9% 1.5%
22

• These numbers suggest that a fully funded system is better than a
pay-as-you-go system.
• Note that a fully funded system can either be public or private.
• Potential drawbacks:
1. Management costs.
2. Distribution of intergenerational risk.
3. Bad choices of households.
4. Lack of redistribution.
23

Possibilities of Reform
• Should we reform the system? Transition!
• Problem: one missing generation: at the introduction of the system
there was one generation that received social security but never paid
taxes.
• Dilemma:
1. Currently young pay double, or
2. Default on the promises for the old, or
3. Increase government debt, ﬁnanced by higher taxes in the future,
i.e. by currently young and future generations.
24

Does Pay-as-you-go Social Security Decrease Saving?
• Without social security saving was given as s = βy1
1+β.
• With social security saving it is given by s =
β(1−τ)y1− SS
1+r
1+β .
• Obviously private saving falls. The social security system as part of the
government does not save, it pays all the tax receipts out immediately
as pensions.
• Hence saving unambiguously goes down with pay-as-you go social se-
curity.
25

Application of the Theory II: Ricardian Equivalence
• What are the effects of government deficits in the economy?
• A first answer: none (Ricardo, 1817, and Barro, 1974).
• How can this be?
• The answer outside our small model is tricky.
26

• Lump-sum taxes.
• Government budget constraints:
G1 = T1 + B
G2 + (1 + r) B = T2
• Consolidating:
G1 +
G2
1 + r
= T1 +
T2
1 + r
• Note that r is constant (you should not worry too much about this).
27

• Original problem:
max
c1,c2
U(c1) + βU(c2)
s.t. c1 +
c2
1 + r
+ T1 +
T2
1 + r
= I
• Now suppose that the government changes timing of taxes T0
1,T0
2 and
government consumption G0
1, G0
2.
28

• Then the problem of the household is:
max
c1,c2
U(c1) + βU(c2)
s.t. c1 +
c2
1 + r
+ T0
1 +
T0
2
1 + r
= I
• Since these new taxes must satisfy:
T1 +
T2
1 + r
= T0
1 +
T0
2
1 + r
problem of the consumer is equivalent!!!
29

Empirical Evidence
• Taxes in the world are not lump-sum.
• Does the Ricardian Equivalence hold?
• Important debate.
30

Dynamic General Equilibrium
1

The Basic Model
• We learned how to think about a household that makes dynamic de-
cisions.
• We learned how to think about a household that makes decisions under
uncertainty.
• We learned how to think about the intertemporal choice of govern-
ment.
• Now we want to introduce investment and put everything together in
a deﬁnition of Dynamic General Equilibrium.
2

Basic Model
• Simple model:
1. Two periods
2. No uncertainty.
• Households can invest in capital.
• Capital tomorrow is:
k2 = (1 − δ) k1 + i1
3

Budget Constraints
• The household budget constrain for the ﬁrst period, given T1, is:
c1 + i1 + b + T1 = w1l1 + r1k1
• Using the fact that k2 = (1 − δ) k1 + i1:
c1 + k2 + b + T1 = w1l1 + r1k1 + (1 − δ) k1
• Budget Constraint in the second period:
c2 + T2 = w2l2 + r2k2 + R2b + (1 − δ) k2
4

Household Problem
max u (c1, 1 − l1) + βu (c2, 1 − l2)
c1 + k2 + b + T1 = w1l1 + r1k1 + (1 − δ) k1
c2 + T2 = w2l2 + r2k2 + R2b + (1 − δ) k2
k2 > 0
5

Two Questions
• Why k2 > 0?
— We have a model with just one agent.
— Capital at the end of second period.
• Why are the households taking the investment decisions and not the
ﬁrms? Role of complete markets.
6

Solving the Household Problem I
• We want to solve for c1, l1, c2, l2, k2 and b given T1, T2, k1, w1, w2,
r1, r2 and R2.
• This was your homework for a parametric example.
7

Solving the Household Problem II
• We build a Lagrangian function:
L = u (c1, 1 − l1) + βu (c2, 1 − l2)
+λ1 (w1l1 + r1k1 + (1 − δ) k1 − c1 − k2 − b − T1)
+λ2 (w2l2 + r2k2 + (1 − δ) k2 + R2b − c1 − k2 − b − T1)
• We take partial derivatives w.r.t. c1, l1, c2, l2, λ1 and λ2, make them
equal to zero and solve the associated system of equations.
8

Solving the Household Problem III
• After going through the previous steps, we have three optimality con-
ditions:
uc (c1, 1 − l1) = β (1 + r2 − δ) uc (c2, 1 − l2)
u1−l (c1, 1 − l1)
uc (c1, 1 − l1)
= w1
u1−l (c2, 1 − l2)
uc (c2, 1 − l2)
= w2
• and one arbitrage condition:
R2 = 1 + r2 − δ
9

Problem of the Firm I
• The problem of the firm is still static.
• In the first period, wants to maximize profits given r1 and w1:
π = Akα
1 l1−α
1 − r1k1 − w1l1
αAkα−1
1 l1−α
1 = r1
(1 − α) Akα
1 l−α
1 = w1
10

Problem of the Firm II
• In the second period, wants to maximize proﬁts given r2 and w2:
π = Akα
2 l1−α
2 − r2k2 − w2l2
αAkα−1
2 l1−α
2 = r2
(1 − α) Akα
2 l−α
2 = w2
11

Government and Market Clearing
• Taxes, T1 and T2 and expenditures, G1 and G2 are given.
• Then, the government budget constraint is:
G1 = T1 + b
G2 + R2b = T2
• Market clearing:
c1 + k2 + G1 = Akα
1 l1−α
1 + (1 − δ) k1
c2 + G2 = Akα
2 l1−α
2 + (1 − δ) k2
12

A Competitive Equilibrium is an allocation {c1, l1, c2, l2, k2, b}, a price sys-
tem {w1, w2, r1, r2, R2} and a government policy {T1, T2, G1, G2} s.t:
1. Given the price system, the government policy and k1 households
choose {c1, l1, c2, l2, k2, b} to maximize their utility.
2. Given the price system and the policy, firms maximize profits.
3. Government satisfies its budget constraint.
4. Markets clear.
13

Extensions of the Model I
• More periods. In fact why not infinite?
• Problem of the household
max
∞X
t=0
βtu (ct, 1 − lt)
ct + kt+1 + bt+1 + Tt = wtlt + rtkt + Rtbt + (1 − δ) kt, ∀ t > 0
• Problem of the firm: maximize profits given rt and wt:
π = Atkα
t l1−α
t − rtkt − wtlt
14

• Budget constraint of the government:
Gt + Rtbt = Tt
• Market clearing:
ct + kt + Gt = Akα
t l1−α
t + (1 − δ) kt
• Transversality conditions (No-Ponzi schemes):
lim
t→∞
βtkt = 0
lim
t→∞
³
Π∞
j=1Rj
´−1
bt = 0
15

A Competitive Equilibrium is an allocation {ct, lt, kt, bt}∞
t=0, a price system
{wt, rt, Rt}∞
t=0 and a government policy {Tt, Gt}∞
t=0 s.t:
1. Given the price system, the government policy and k0 households
choose {ct, lt, bt}∞
t=0 to maximize their utility.
2. Given the price system and the policy, ﬁrms maximize proﬁts.
4. Markets clear:
16

• Proof of existence of equilibrium.
• The Welfare theorems hold.
• How do we ﬁnd the equilibrium? Dynamic Programing.
17

Visiting Old Friends
• This model with inﬁnite period is an old friend of ours: Solow Model
with endogenous labor supply and savings.
• Nearly everything we learned in the Growth part of the class will hold
here (convergence, steady states and BGPs, transitional dynamics...).
• Also, if we make A endogenous, we will have an Endogenous Growth
Model with endogenous labor supply and savings.
18

Extensions of the Model II: Uncertainty
• Problem of the household:
max
∞X
t=0
X
st∈St
βtπ
³
st
´
u(c
³
st
´
, l
³
st
´
)
s.t. c
³
st
´
+ k
³
st
´
+
X
st+1∈S
³
q
³
st, st+1
´
a
³
st, st+1
´
+ b
³
st, st+1
´´
=
w
³
st
´
l
³
st
´
+ r
³
st
´
k
³
st−1
´
+ (1 − δ) k
³
st−1
´
+
+R
³
st−1, st
´
b
³
st−1, st
´
+ a
³
st−1, st
´
19

• Problem of the firm: maximize profits given r
³
st
´
and w
³
st
´
:
π
³
st
´
= A
³
st
´
k
³
st−1
´α
l
³
st
´1−α
− r
³
st
´
k
³
st−1
´
− w
³
st
´
l
³
st
´
• Role of A
³
st
´
.
• Government:
n
T
³
st
´
, G
³
st
ó∞
t=0
20

Equilibrium
A S.M. equilibrium is an allocation
n
c∗
³
st
´
, l∗
³
st
´
, k∗
³
st
ó∞
t=0,st∈St ,
portfolio decisions
n
a∗
³
st, st+1
´
, b∗
³
st, st+1
ó∞
t=0,st∈St and prices
n
q∗
³
st, st+1
´
, R∗
³
st, st+1
´
, w∗
³
st
´
, r∗
³
st
ó∞
t=0,st∈St such that:
1. Given prices, the allocation solves the problem of the consumer and of
the firm.
3. Markets clear.
21

Further Extensions of the Model
• No-lump taxes.
• Money.
• Life Cycle: OLG
• Diﬀerent market imperfections.
22

Money
1

Why Money?
• Two questions:
1. Modern economies use money. Why?
2. Changes in the amount of money can aﬀect nominal and real vari-
ables in the economy.
• It is important to answer this questions in order to implement monetary
policy.
2

Uses of Money
• Unit of account: contracts are usually denominated in terms of money.
• Store of Value: money allows consumers to trade current goods for
future goods.
• Medium of Exchange.
3

Other Objects
• Other objects, like stocks and bonds, can be store of value and medium
of exchange.
• Moreover, as store of value, stocks and bonds are better than money,
since they give a positive rate of return.
• However, stocks and bonds are not very eﬃcient as a medium of
exchange because:
1. Agents are not usually well-informed about the exact value of
stocks.
2. It is not always easy to sell these assets.
4

• Hence, what distinguishes money is that it plays in a very eﬃcient way
its role as a medium of exchange.
• There is a large literature that deals with the role of money as a
medium of exchange.
• In absence of regular money, other obsjects appear as mediums of
exchage (cigarettes in POW’s camps).
5

Two classes of Models
• “Deep Models”
Explicitly model the fundamental reason money is used a medium of
exchange, i.e. the existence of frictions in trade.
• “Applied Models”
Simply assume that money has to be used to carry out some transac-
tions and proceed from there.
6

Deep Models
• Kiyotaki and Wrigth (1989).
• The main reason for money to have value is the double-coincidence of
wants problem. In a specialized economy is not easy to ﬁnd someone
that has what you like and, at the same time, likes what you have.
• Money reduces this problem by making exchange possible in a single-
coincidence of wants meeting.
7

Applied Models
• We are going to focus in the more applied perspective on money.
• We will simply assume that there are some goods that only money can
buy (Cash Goods): “Cash-in-Advance Models”.
• Other ways to do it: “Money in the Utility function”.
• You can show both approaches are equivalent.
• Are we doing the right thing (Wallace, 2001)?
8

A Simple Monetary Model I
• Two assets: Money, M and Nominal Bonds, B
• Nominal Bond is an asset that sells for one unit of money in the current
period and pays oﬀ 1 + R units of money in the future period.
• R = rate of return on a bond in terms of money (nominal interest
rate).
• r = real interest rate.
9

A Simple Monetary Model II
• π =inﬂation
π =
Ptoday − Pyesterday
Pyesterday
• Notice that now we have a level of nominal prices.
• Then:
1 + r =
1 + R
1 + π
' R − π
10

Household I
• Representative Household consists of a worker and a shopper.
• We assume that Cash Goods can only be acquired if an agent has
money from the previous period.
• Therefore the demand for money is equivalent to the demand for future
Cash Goods.
• The household has to decide the demand for Cash Goods cm
t , Credit
Goods cc
t, Money Mt, Bonds Bt and labor supply lt.
11

Household II
• Utility Function
P∞
t=0 βtu (cm
t , cc
t, 1 − lt)
• Budget constraint
Pt (cm
t + cc
t) + Bt + Mt = Mt−1 + (1 + Rt)Bt−1 + Ptwtlt
• Cash In Advance Constraint:
Ptcm
t ≤ Mt−1
• No-Ponzi Scheme condition:
lim
t→∞
βtu1 (cm
t , cc
t, 1 − lt) Bt = 0
12

Optimality Conditions I
• Lagrangian:
∞X
t=0



βtu (cm
t , cc
t, 1 − l) + λt
Ã
Mt−1 + (1 + Rt)Bt−1 + Ptwtlt
−Pt (cm
t + cc
t) − Bt − Mt
!
+µt (Mt−1 − Ptcm
t )



• First order conditions:
βtu1 (cm
t , cc
t, 1 − lt) = (λt + µt) Pt (1)
βtu2 (cm
t , cc
t, 1 − lt) = λtPt (2)
βtu3 (cm
t , cc
t, 1 − lt) = λtPtwt (3)
−λt−1 + λt + µt = 0 (4)
−λt−1 + λt(1 + Rt) = 0 (5)
13

Optimality Conditions II
• Dividing (2) by (1):
u2 (cm
t , cc
t, 1 − lt)
u1
¡
cm
t , cc
t, 1 − lt
¢ =
λt
λt + µt
• But using (4) and (5):
u2 (cm
t , cc
t, 1 − lt)
u1
¡
cm
t , cc
t, 1 − lt
¢ =
1
1 + Rt
=
1
(1 + πt) (1 + rt)
• Interpretation.
14

Optimality Conditions III
• Labor supply:
u3 (cm
t , cc
t, 1 − lt)
u2
¡
cm
t , cc
t, 1 − lt
¢ = wt
• Then demand for money depends on rt, πt and wt.
• Putting everything in equilibrium. In particular how are rt and wt pind
down?
15

Using the Model
1. Changes in money supply (Mt changes over time): eﬀects on output
and on price level.
2. Changes in wages.
3. Changes in real interest rate.
16

Unemployment
1

Why Unemployment?
• So far we have studied models where labor market clears.
• Is that a good assumption?
• Why is unemployment important?
1. Reduces income
2. Increases inequality.
• How can we think about unemployment in an equilibrium model?
2

Concepts and Facts from the Labor Market
• The labor force is the number of people, 16 or older, that are either
employed or unemployed but actively looking for a job. We denote the
labor force at time t by Nt.
• Note that actively looking for a job is an ambiguous term.
• Let WPt denote the total number of people in the economy that are
of working age (16 - 65) at date t . The labor force participation rate
ft is deﬁned as the fraction of the population in working age that is
in the labor force, i.e. ft = Nt
WPt
.
3

• The number of unemployed people are all people that don’t have a job.
We denote this number by Ut. Similarly we denote the total number
of people with a job by Lt. Obviously Nt = Lt + Ut. We define the
unemployment rate ut by
ut =
Ut
Nt
• The job losing rate bt is the fraction of the people with a job which is
laid off during a particular time, period, say one month (it is crucial
for this definition to state the time horizon). The job finding rate et
is the fraction of unemployed people in a month that find a new job.
4

Basic Facts
• U.S. Labor Force in feb 2002: 142 million people
• U.S. working age population in 2002: 212 million people
• Labor force participation rate of about 67.0%.
• Between 1967 and 1993 the average job losing rate was 2.7% per
month
• Average job ﬁnding rate was 43%.
• Average unemployment rate during this time period was about 6.2%
5

Job Creation and Destruction
• The gross job creation Crt between period t − 1 and t equals the em-
ployment gain summed over all plants that expand or start up between
period t − 1 and t.
• The gross job destruction Drt between period t − 1 and t equals the
employment loss summed over all plants that contract or shut down
between period t − 1 and t.
• The net job creation Nct between period t−1 and t equals Crt −Drt.
• The gross job reallocation Rat between period t − 1 and t equals
Crt + Drt.
6

Main Findings of Davis, Haltiwanger and Schuh (1996)
• Data from all manufacturing plants in the US with 5 or more employees
from 1963 to 1987. In the years they have data available, there were
between 300,000 and 400,000 plants.
• Gross job creation Crt and job destruction Drt are remarkably large.
In a typical year 1 out of every ten jobs in manufacturing is destroyed
and a comparable number of jobs is created at diﬀerent plants.
• Most of the job creation and destruction reﬂects highly persistent
plant-level employment changes. Most jobs that vanish at a particular
plant fail to reopen at the same location within the next two years.
7

• Job creation and destruction are concentrated at plants that experi-
ence large percentage employment changes. Two-thirds of job cre-
ation and destruction takes place at plants that expand or contract by
25% or more within a twelve-month period. About one quarter of job
destruction takes place at plants that shut down.
• Job destruction exhibits greater cyclical variation than job creation.
In particular, recessions are characterized by a sharp increase in job
destruction accompanied by a mild slowdown in job creation.
8

Unemployment and the Business Cycle
• Gross job creation is relatively stable over the business cycle, whereas
gross job destruction moves strongly countercyclical: it is high in re-
cessions and low in booms.
• In severe recessions such as the 74-75 recession or the 80-82 back to
back recessions up to 25% of all manufacturing jobs are destroyed
within one year, whereas in booms the number is below 5%.
• Time a worker spends being unemployed also varies over the business
cycle, with unemployment spells being longer on average in recession
years than in years before a recession.
9

• Length of unemployment spells:
Unemployment Spell 1989 1992
< 5 weeks 49% 35%
5 - 14 weeks 30% 29%
15 - 26 weeks 11% 15%
> 26 weeks 10% 21%
• Other countries: in Germany, France or the Netherlands about two
thirds of all unemployed workers in 1989 were unemployed for longer
than six months!!
10

The Evolution of the Unemployment Rate
• Ut = Number of unemployed at t
• Nt = Labor Force in t
• Lt = Nt − Ut = Number of employed in t
• ut = Ut
Nt
= unemployment rate
• s = job losing rate
11

• e = job ﬁnding rate
• Assume that Nt = (1 + n)Nt−1
• Then we have
Ut = (1 − e)Ut−1 + sLt−1
= (1 − e)Ut−1 + s(Nt−1 − Ut−1)
12

• Dividing both sides by Nt = (1 + n)Nt−1 yields
ut =
Ut
Nt
=
(1 − e)Ut−1
(1 + n)Nt−1
+
s(Nt−1 − Ut−1)
(1 + n)Nt−1
=
1 − e
1 + n
ut−1 +
s(1 − ut−1)
1 + n
=
1 − e − s
1 + n
ut−1 +
s
1 + n
13

Steady State Rate of Unemployment
• In theory: steady state unemployment rate, absent changes in n, s, e
• Some people call it “Natural Rate”: example of theory ahead of lan-
guage.
• Origin of the idea: Friedman 1969
14

• Solve for u∗ = ut−1 = ut
u∗ =
1 − e − s
1 + n
u∗ +
s
1 + n
n + e + s
1 + n
u∗ =
s
1 + n
u∗ =
s
n + e + s
• From data s = 2.7%, e = 43% and n = 0.09%
• u∗ = 5.9%
15

Determinants of the Rate of Unemployment
• We just presented an accounting exercise.
• There was no theory on it.
• We want to have a model to think about the diﬀerent elements of the
model (b, e, etc.).
16

Several Models to Think about Unemployment
• Search Model.
• Eﬃciency Wages Models.
• Sticky Wages and Prices Models.
17

Basic Search Model I
1. Matching is costly. Think about getting a date.
2. We can bring our intuition to the job market.
3. Contribution of McCall (1970).
18

Basic Search Model II
1. Ve (w): utility from being employed at wage w.
2. Vu: utility from being unemployed. Will depend on size of unemploy-
ment beneﬁt.
3. Workers get random job oﬀers at wage wi with some probability p.
4. Reservation Wage: wage such that Vu = Ve (w∗).
5. Rule: accept the job if wi > w∗. Otherwise reject.
19

Basic Search Model III
Finding the unemployment rate in the Search Model:
1. Take a separation rate s. Then ﬂow of workers from employment into
unemployment is s (1 − U).
2. H (w): fraction of unemployees that get job oﬀers s.t. wi > w.
3. Then new employees are UpH (w∗) .
4. U∗ is such that U∗pH (w∗) = s (1 − U∗) or
U∗ =
s
s + pH (w∗)
=
s
s + pH (Vu)
20

Basic Search Model IV
Determinants of Unemployment Rate in the Search Model:
1. Unemployment Insurance: Length and generosity of unemployment
insurance vary greatly across countries. US replacement rate is 34%.
Germany, France and Italy the replacement rate is about 67%, with
duration well beyond the first year of unemployment.
2. Minimum Wages: If the minimum wage is so high that it makes certain
jobs unprofitable, less jobs are offered and job finding rates decline.
21

Efficiency Wage Model I
• Asymmetric Information is a common factor in labor markets
• In particular monitor effort of employee is tough.
• Firms will try to induce workers to put more effort.
• How can this create unemployment?
• Shaphiro-Stiglitz (1984) model.
22

Efficiency Wage Model II
• The economy consist of a large number of workers L∗ and of firms N.
• The worker can supply two levels of effort, e = 0 or e = e∗.
• Utility of the worker
u =
(
w − e if employed
0 if unemployed
• Household has a discount rate ρ (in our usual notation ρ ' 1 − β).
23

Efficiency Wage Model III
The worker can be in three different states:
1. E: employed and exerting effort e∗.
2. S: employed and shirking, e = 0.
3. U: unemployed.
24

Efficiency Wage Model IV
• If the worker exerts effort, he will lose his job with probability b.
• If the worker shirks, he will lose his job with probability q + b.
• If the worker is unemployed, it finds a job with probability a.
25

Eﬃciency Wage Model V
• Then, the value of being unemployed is
ρVE = w − e∗ − b (VE − VU)
• The value of shirking is
ρVS = w − (b + q) (VS − VU)
• And the value of unemployment is:
ρVU = a (VE − VU)
26

Eﬃciency Wage Model VI
• Firm’s problem:Then, the value of being unemployed is
max
w,L+S
π = F(e∗L) − w (L + S)
where F0 > 0 and F00 < 0.
• We will assume
F0
Ã
e∗L∗
N
!
> 1
27

The No-Shirking Condition I
The ﬁrm will pick w to satisfy:
VE = VS
or:
w − e∗ − b (VE − VU) = w − (b + q) (VS − VU)
e∗ + b (VE − VU) = (b + q) (VS − VU)
e∗ + b (VE − VU) = (b + q) (VE − VU)
VE − VU =
e∗
q
28

The No-Shirking Condition II
Also:
ρVE = w − e∗ − b (VE − VU)
ρVU = a (VE − VU)
Substituting
ρ(VE − VU) = w − e∗ − (a + b) (VE − VU)
Using VE − VU = e∗
q
ρ(
e∗
q
) = w − e∗ − (a + b)
Ã
e∗
q
!
or
w = e∗ + (a + b + ρ)
Ã
e∗
q
!
29

The No-Shirking Condition II
Now, note that in an steady state
NLb = (L∗ − NL)a
or:
a =
NLb
L∗ − NL
and
a + b =
NLb
L∗ − NL
+ b =
NLb + bL∗ − NLb
L∗ − NL
=
bL∗
L∗ − NL
so we get:
w = e∗ +
Ã
bL∗
L∗ − NL
+ ρ
! Ã
e∗
q
!
This equation is called the No-Shirking Condition.
30

Closing the Model
• If VE = VS, all the workers will exert effort and the firm’s profits will
be:
F(e∗L) − wL
with FOC:
e∗F0(e∗L) = w
• Since w = e∗ +
³
bL∗
L∗−NL + p
´ ³
e∗
q
´
, we have:
e∗F0(e∗L) = e∗ +
Ã
bL∗
L∗ − NL
+ ρ
! Ã
e∗
q
!
z
31

w
NL
e*
L*
L
d
NSC
E
The Shapiro-Stiglitz Model

International Labor Market Comparisons I
Unemployment and Long-Term Unemployment in OECD
Unemployment (%) ≥ 6 Months ≥ 1 Year
74-9 80-9 95 79 89 95 79 89 95
Belgium 6.3 10.8 13.0 74.9 87.5 77.7 58.0 76.3 62.4
France 4.5 9.0 11.6 55.1 63.7 68.9 30.3 43.9 45.6
Germany 3.2 5.9 9.4 39.9 66.7 65.4 19.9 49.0 48.3
Netherlands 4.9 9.7 7.1 49.3 66.1 74.4 27.1 49.9 43.2
Spain 5.2 17.5 22.9 51.6 72.7 72.2 27.5 58.5 56.5
Sweden 1.9 2.5 7.7 19.6 18.4 35.2 6.8 6.5 15.7
UK 5.0 10.0 8.2 39.7 57.2 60.7 24.5 40.8 43.5
US 6.7 7.2 5.6 8.8 9.9 17.3 4.2 5.7 9.7
OECD Eur. 4.7 9.2 10.3 - - - 31.5 52.8 -
Tot. OECD 4.9 7.3 7.6 - - - 26.6 33.7 -
32

International Labor Market Comparisons II
Distribution of Long-Term Unemployment (longer than 1 year) by Age in
1990
Age Group
15 − 24 25 − 44 ≥ 45
Belgium 17 62 20
France 13 63 23
Germany 8 43 48
Netherlands 13 64 23
Spain 34 38 28
Sweden 9 24 67
UK 18 43 39
US 14 53 33
33

International Labor Market Comparisons III
Net Unemployment Replacement Rate for Single-Earner Households
Single With Dependent Spouse
1. Y. 2.-3. Y. 4.-5. Y. 1. Y. 2.-3. Y. 4.-5. Y.
Belgium 79 55 55 70 64 64
France 79 63 61 80 62 60
Germany 66 63 63 74 72 72
Netherlands 79 78 73 90 88 85
Spain 69 54 32 70 55 39
Sweden 81 76 75 81 100 101
UK 64 64 64 75 74 74
US 34 9 9 38 14 14
34

How can we use theory to think about these facts?
The Explanations:
1. Rigidities in the labor market
2. Wage bargaining
3. Welfare state
4. International trade
5. A restrictive economic policy
35

Rigidities in the labor market: Hopenhayn and Rogerson
1. Firing restrictions
2. Work arrangements
3. Minimum wages
36

Wage bargaining: Cole and Ohanian, Caballero and Hammour
1. Eﬀect of “insiders” vs. “outsiders”.
2. Trade Unions.
37

Welfare State: Ljungvist and Sargent
1. High welfare beneﬁts
2. Rapidly changing economy
38

International Trade
1. Increase in trade during last 25 years.
2. Basic Trade Model Implications.
3. Facts do not agree with theory
39

Possible Policy Solutions
• Reform Labor Market Institutions.
• Reduce Trade Unions power.
• Changing Unemployment Insurance.
• Reform Educational Systems.
• Increasing Mobility.
Think about Political Economy.
40

Business Cycles
1

Business Cycles
• U.S. economy ﬂuctuates over time.
• How can we build models to think about it?
• Do we need diﬀerent models than before to do so? Traditionally the
answer was yes. Nowadays the answer is no.
2

Business Cycles and Economic Growth
• How diﬀerent are long-run growth and the business cycle?
Changes in Output per Worker Secular Growth Business Cycle
Due to changes in capital 1/3 0
Due to changes in labor 0 2/3
Due to changes in productivity 2/3 1/3
• We want to use the same models with a slightly diﬀerent focus.
3

Representing the Business Cycle I
• How do we look at the data?
• What is the cyclical component of a series and what is the trend?
• Is there a unique way to decide it?
4

Representing the Business Cycle II
• Suppose that we have T observations of the stochastic process X,
{xt}t=∞
t=1 .
• Hodrick-Prescott (HP) ﬁlter decomposes the observations into the
sum of a trend component, xt
t and a cyclical component xc
t:
xt = xt
t + xc
t
5

Representing the Business Cycle III
• How? Solve:
min
xt
t
TX
t=1
³
xt − xt
t
´2
+ λ
T−1X
t=2
h³
xt
t+1 − xt
t
´
−
³
xt
t − xt
t−1
´i2
(1)
• Intuition.
• Meaning of λ:
1. λ = 0 ⇒trivial solution (xt
t = xt).
2. λ = ∞ ⇒linear trend.
6

Representing the Business Cycle IV
• To compute the HP ﬁlter is easier to use matricial notation, and rewrite
(1) as:
min
xt
³
x − xt
´0 ³
x − xt
´
+ λ
³
Axt
´0 ³
Axt
´
where x = (x1, ..., xT )0, xt =
³
xt
1, ..., xt
T
´0
and:
A =








1 −2 1 0 · · · · · · 0
0 1 −2 1 · · · · · · 0
... ... ... ... ... ... ...
... ... ... 1 −2 1 0
0 · · · · · · · · · 1 −2 1








(T−2)×T
7

Representing the Business Cycle V
• First order condition:
xt − x + λA0Axt = 0
or
xt =
³
I + λA0A
´−1
x
•
¡
I + λA0A
¢−1
is a sparse matrix (with density factor (5T − 6) /T2).
We can exploit this property.
8

.
Figure 1: Idealized Business Cycles

Figure 2: Percentage Deviations from Trend in
Real GDP from 1947 to 1999

Figure 3: Time Series Plots of x and y

.
Figure 4: Correlations Between Variables
y and x

Figure 5: Leading and Lagging Variables

Figure 6: Percentage Deviations from Trend in Real GDP
(colored line) and the Index of Leading Economic
Indicators (black line) for the Period 1959-1999

Figure 7: Percentage Deviations from Trend in Real
Consumption (black line) and Real GDP (colored line)

Figure 8: Percentage Deviations from Trend in Real
Investment (black line) and Real GDP (colored line)

Figure 9: Scatter Plot for the Percentage Deviations from
Trend in the Price Level (the Implicit GDP Price Deflator)
and Real GDP

.
Figure 10: Price Level and GDP

Figure 11: Percentage Deviations from Trend in the Money
Supply (black line) and Real GDP (colored line)
for the Period 1947-1999

Figure 12: Percentage Deviations from Trend in
Employment (black line) and Real GDP (colored line)

Real Business Cycles
1

Fluctuations
• Economy ﬂuctuates over time.
• Is there a systematic phenomenon we need to explain?
• Simple random walk:
yt = yt−1 + εt
εt ∼ N(0, σ)
• Is output well described by a Random Walk? Plosser and Nelson
(1992) and Unit Root testing.
2

0 20 40 60 80 100 120 140 160 180 200
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
A Random Walk

Fluctuations in Equilibrium
• We want to think about business cycles using an equilibrium perspec-
tive.
• Traditionally economist did not use an equilibrium approach to address
this issue.
• Big innovation: Lucas (1972).
• How can we generate ﬂuctuations in equilibrium?
3

Real Business Cycles
• We learned how to map preferences (for the household), technology
(for the ﬁrm) and a government policy into a Competitive Equilibrium.
• If we let preferences, technology or the government preferences change
over time, the equilibrium sequence will also ﬂuctuate.
• All these (preferences, technology, policy) are real factors (as opposed
to monetary).
• This is the reason we call this approach Real Business Cycles.
• Big innovation: Kydland and Prescott (1982).
4

Intuition I
• Let us go back to our Robinson Crusoe’s Economy.
• How will Robinson do if he wakes up and today is a sunny day?
• And if it is rainy?
• Basic idea: intertemporal substitution.
5

Intuition II
• We will have an initial shock: change in preferences, technology or
policy.
• Then we will have a propagation mechanism: intertemporal labor sub-
stitution and capital accumulation.
• We will have ﬂuctuations as an equilibrium outcome.
6

Productivity Shocks
• We will study ﬁrst the eﬀects of changes to technology.
• Remember that:
.
Y
Y
=
.
A
A
+ α
.
K
K
+ (1 − α)
.
L
L
and that the Solow Residual is:
.
A
A
=
.
Y
Y
− α
.
K
K
− (1 − α)
.
L
L
• How do the Solow Residual and GDP move together?
7

Question
• Let us suppose that we have an economy that is hit over time by
productivity shocks with the same characteristics that the ones that
hit the US economy.
• How does this economy behave over time?
• In particular, how do the variances and covariances of the main vari-
ables in our economy compare with those observed in the US economy?
8

Household Problem
• Preferences:
max E
∞X
t=0
β {log ct + ψ log (1 − lt)}
• Budget constraint:
ct + kt+1 = wtlt + rtkt + (1 − δ) kt, ∀ t > 0
9

Problem of the Firm
• Neoclassical production function:
yt = kα
t (eztlt)1−α
• By proﬁt maximization:
αkα−1
t (eztlt)1−α
= rt
(1 − α) kα
t (eztlt)1−α
l−1
t = wt
10

Evolution of the technology
• zt changes over time.
• It follows the AR(1) process:
zt = ρzt−1 + εt
εt ∼ N(0, σ)
• Interpretation of ρ.
11

• We can deﬁne a competitive equilibrium in the standard way.
• The competitive equilibrium is unique.
• This economy satisﬁes the conditions that assure that both welfare
theorems hold.
• Why is this important? We could solve instead the Social Planner’s
Problem associated with it.
• Advantages and disadvantages of solving the social planner’s problem.
12

The Social Planner’s Problem
• It has the form:
max E
∞X
t=0
β {log ct + ψ log (1 − lt)}
ct + kt+1 = kα
t (eztlt)1−α
+ (1 − δ) kt, ∀ t > 0
zt = ρzt−1 + εt, εt ∼ N(0, σ)
• This is a dynamic optimization problem.
13

Computing the RBC
• The previous problem does not have a known “paper and pencil” so-
lution.
• We will work with an approximation: Perturbation Theory.
• We will undertake a ﬁrst order perturbation of the model.
• How well will the approximation work?
14

Equilibrium Conditions
From the household problem+ﬁrms’s problem+aggregate conditions:
1
ct
= βEt
(
1
ct+1
³
1 + αkα−1
t (eztlt)1−α
− δ
´
)
ψ
ct
1 − lt
= (1 − α) kα
t (eztlt)1−α
l−1
t
ct + kt+1 = kα
t (eztlt)1−α
+ (1 − δ) kt
zt = ρzt−1 + εt
15

Finding a Deterministic Solution
• We search for the ﬁrst component of the solution.
• If σ = 0, the equilibrium conditions are:
1
ct
= β
1
ct+1
³
1 + αkα−1
t l1−α
t − δ
´
ψ
ct
1 − lt
= (1 − α) kα
t l−α
t
ct + kt+1 = kα
t l1−α
t + (1 − δ) kt
16

Deterministic Steady State
• The equilibrium conditions imply a steady state:
1
c
= β
1
c
³
1 + αkα−1l1−α − δ
´
ψ
c
1 − l
= (1 − α) kαl−α
c + δk = kαl1−α
• Or simplifying:
1
β
= 1 + αkα−1l1−α − δ
ψ
c
1 − l
= (1 − α) kαl−α
c + δk = kαl1−α
17

Solving the Steady State
Solution:
k =
µ
Ω + ϕµ
l = ϕk
c = Ωk
y = kαl1−α
where ϕ =
³
1
α
³
1
β − 1 + δ
´´ 1
1−α
, Ω = ϕ1−α − δ and µ = 1
ψ (1 − α) ϕ−α.
18

Linearization I
• Loglinearization or linearization?
• Advantages and disadvantages
• We can linearize and perform later a change of variables.
19

Linearization II
We linearize:
1
ct
= βEt
(
1
ct+1
³
1 + αkα−1
t (eztlt)1−α
− δ
´
)
ψ
ct
1 − lt
= (1 − α) kα
t (eztlt)1−α
l−1
t
ct + kt+1 = kα
t (eztlt)1−α
+ (1 − δ) kt
zt = ρzt−1 + εt
around l, k, and c with a First-order Taylor Expansion.
20

Linearization III
We get:
−
1
c
(ct − c) = Et
(
−1
c (ct+1 − c) + α (1 − α) βy
kzt+1+
α (α − 1) β y
k2 (kt+1 − k) + α (1 − α) β y
kl (lt+1 − l)
)
1
c
(ct − c) +
1
(1 − l)
(lt − l) = (1 − α) zt +
α
k
(kt − k) −
α
l
(lt − l)
(ct − c) + (kt+1 − k) =



y
µ
(1 − α) zt + α
k (kt − k) +
(1−α)
l (lt − l)
¶
+ (1 − δ) (kt − k)



zt = ρzt−1 + εt
21

Rewriting the System I
Or:
α1 (ct − c) = Et {α1 (ct+1 − c) + α2zt+1 + α3 (kt+1 − k) + α4 (lt+1 − l)}
(ct − c) = α5zt +
α
k
c (kt − k) + α6 (lt − l)
(ct − c) + (kt+1 − k) = α7zt + α8 (kt − k) + α9 (lt − l)
zt = ρzt−1 + εt
22

Rewriting the System II
where
α1 = −1
c α2 = α (1 − α) βy
k
α3 = α (α − 1) β y
k2 α4 = α (1 − α) β y
kl
α5 = (1 − α) c α6 = −
µ
α
l + 1
(1−l)
¶
c
α7 = (1 − α) y α8 = yα
k + (1 − δ)
α9 = y
(1−α)
l y = kαl1−α
23

Rewriting the System III
After some algebra the system is reduced to:
A (kt+1 − k) + B (kt − k) + C (lt − l) + Dzt = 0
Et (G (kt+1 − k) + H (kt − k) + J (lt+1 − l) + K (lt − l) + Lzt+1 + Mzt) = 0
Etzt+1 = ρzt
24

Guess Policy Functions
We guess policy functions of the form (kt+1 − k) = P (kt − k) + Qzt and
(lt − l) = R (kt − k) + Szt, plug them in and get:
A (P (kt − k) + Qzt) + B (kt − k)
+C (R (kt − k) + Szt) + Dzt = 0
G (P (kt − k) + Qzt) + H (kt − k) + J (R (P (kt − k) + Qzt) + SNzt)
+K (R (kt − k) + Szt) + (LN + M) zt = 0
25

Solving the System I
Since these equations need to hold for any value (kt+1 − k) or zt we need
to equate each coeﬃcient to zero, on (kt − k):
AP + B + CR = 0
GP + H + JRP + KR = 0
and on zt:
AQ + CS + D = 0
(G + JR) Q + JSN + KS + LN + M = 0
26

Solving the System II
• We have a system of four equations on four unknowns.
• To solve it note that R = − 1
C (AP + B) = − 1
CAP − 1
CB
• Then:
P2 +
µ
B
A
+
K
J
−
GC
JA
¶
P +
KB − HC
JA
= 0
a quadratic equation on P.
27

Solving the System III
• We have two solutions:
P = −
1
2

−
B
A
−
K
J
+
GC
JA
±
Ãµ
B
A
+
K
J
−
GC
JA
¶2
− 4
KB − HC
JA
!0.5


one stable and another unstable.
• If we pick the stable root and ﬁnd R = − 1
C (AP + B) we have to a
system of two linear equations on two unknowns with solution:
Q =
−D (JN + K) + CLN + CM
AJN + AK − CG − CJR
S =
−ALN − AM + DG + DJR
AJN + AK − CG − CJR
28

Calibration
• What does it mean to calibrate a model?
• Our choices
Calibrated Parameters
Parameter β ψ α δ ρ σ
Value 0.99 1.75 0.33 0.023 0.95 0.01
29

Computation I
• In practice you do all that in the computer.
• One of the main tools of modern macroeconomic theory is the com-
puter.
• We build small laboratory economies inside our computers and we run
experiments on them.
• If our economy behaves well in those experiments we know the answer,
we are conﬁdent about its answers for questions we do not know.
• Modern macroeconomics is a Quantitative Science.
30

Computation II
• I use some Matlab code: rbc.mod for more sophisticated manipulation.
• How does it work?
• If any of you is thinking about graduate studies in economics, you
should learn a programming language NOW (Matlab, Fortran 90,
C++).
31

Solution I
Basic results are easy to understand:
• You want to work hard when productivity is high.
• You want to save more when productivity is high. Relation with con-
sumption smoothing.
• Reverse eﬀects happen when productivity is low.
32

Solution II
• We have an initial shock: productivity changes.
• We have a transmission mechanism: intertemporal substitution and
capital accumulation.
• We can look at a simulation from this economy.
• Why only a simulation?
33

Comparison with US economy
• Simulated Economy output ﬂuctuations are around 75% as big as
observed ﬂuctuations.
• Consumption is less volatile than output.
• Investment is much more volatile.
• Behavior of hours.
34

Assessment of the Basic Real Business Model
• It accounts for a substantial amount of the observed ﬂuctuations.
• It accounts for the covariances among a number of variables.
• It has some problems accounting for the behavior of the hours worked.
• More important question: where do productivity shocks come from?
35

Negative Productivity Shocks I
• The model implies that half of the quarters we have negative technol-
ogy shocks.
• Is this plausible? What is a negative productivity shocks?
• Role of trend.
36

Negative Productivity Shocks II
• s.d. of shocks is 0.01. Mean quarter productivity growth is 0.0048 (to
give us a 1.9% growth per year).
• As a consequence, we would only observe negative technological shocks
when εt < −0.0048.
• This happens in the model around 33% of times.
• Ways to ﬁx it.
37

Some Policy Implications
• The basic model is Pareto-efficient.
• Fluctuations are the optimal response to a changing environment.
• Fluctuations are not a sufficient condition for inefficiencies or for gov-
ernment intervention.
• In fact in this model the government can only worsen the allocation.
• Recessions have a “cleansing” effect.
38

Extensions
• We can extend our model in several directions.
• Fiscal Policy shocks (McGrattan, 1994).
• Agents with Finite Lives (R´ıos-Rull, 1996).
• Indivisible Labor (Rogerson, 1988, and Hansen, 1985).
• Home Production (Benhabib, Rogerson and Wright, 1991).
• Money (Cooley and Hansen, 1989).
39

Fiscal Policy
• We can use the model with taxes and public spending to think about
fiscal policy.
• Issue non trivial: we are not in a Ricardian Equivalence world.
• Two different questions:
1. Effects of the fiscal policy on the economy.
2. What is the optimal fiscal policy over the cycle.
40

Eﬀects of Fiscal Policy
• Basic Intuition.
• The model generates:
1. a positive correlation between government spending and output.
2. a positive correlation between temporarily lower taxes and output.
• How can we use our model to think about Bush’s tax plan?
41

Optimal Fiscal Policy
• How do we want to conduct ﬁscal policy over the cycle?
• Remember that this is the Ramsey Problem.
• Chari, Christiano, and Kehoe (1994):
1. No tax on capital (Chamley, 1986, Judd, 1985).
2. Tax labor smoothly.
3. Use debt as a shock absorber.
• Evaluating Bush’s tax plan from a Ramsey perspective.
42

Time Inconsistency I
• Previous analysis imply that governments can commit themselves.
• Is this a realistic assumption?
• Time inconsistency problem.
• Original contribution by Kydland and Prescott (1977).
• Main example: tax on capital
43

Time Inconsistency II
• What can we say about ﬁscal policy in this situations?
• Use the tools of game theory.
• Main contributions:
1. Chari and Kehoe (1989).
2. Klein and R´ıos-Rull (2001).
3. Stacchetti and Phelan (2001).
44

Putting our Theory to Work: the Great Depression
• Great Depression is a unique event in US history.
• Timing 1929-1933.
• Major changes in the US Economic policy: New Deal.
• Can we use the theory to think about it?
45

Data on the Great Depression
Year ur Y C I G i π
1929 3.2 203.6 139.6 40.4 22.0 5.9 −
1930 8.9 183.5 130.4 27.4 24.3 3.6 −2.6
1931 16.3 169.5 126.1 16.8 25.4 2.6 −10.1
1932 24.1 144.2 114.8 4.7 24.2 2.7 −9.3
1933 25.2 141.5 112.8 5.3 23.3 1.7 −2.2
1934 22.0 154.3 118.1 9.4 26.6 1.0 7.4
1935 20.3 169.5 125.5 18.0 27.0 0.8 0.9
1936 17.0 193.2 138.4 24.0 31.8 0.8 0.2
1937 14.3 203.2 143.1 29.9 30.8 0.9 4.2
1938 19.1 192.9 140.2 17.0 33.9 0.8 −1.3
1939 17.2 209.4 148.2 24.7 35.2 0.6 −1.6
1940 14.6 227.2 155.7 33.0 36.4 0.6 1.6
46

Output, Inputs and TFP During the Great Depression (Ohanian, 2001)
• Theory
.
A
A
=
.
Y
Y
− α
.
K
K
− (1 − α)
.
L
L
• Data (1929=100)
Year Y L K A
1930 89.6 92.7 102.5 94.2
1931 80.7 83.7 103.2 91.2
1932 66.9 73.3 101.4 83.4
1933 65.3 73.5 98.4 81.9
• Why did TFP fall so much?
47

Potential Reasons
• Changes in Capacity Utilization.
• Changes in Quality of Factor Inputs.
• Changes in Composition of Production.
• Labor Hoarding.
• Increasing Returns to Scale.
48

Other Reasons for Great Depression (Cole and Ohanian 1999)
• Monetary Shocks: Monetary contraction, change in reserve require-
ments too late
• Banking Shocks: Banks that failed too small
• Fiscal Shocks: Government spending did rise (moderately)
• Sticky Nominal Wages: Probably more important for recovery
49

Output and Productivity after the Great Depression (Cole and Ohanian,
2003)
• Data (1929=100); data are detrended
Year Y A
1934 64.4 92.6
1935 67.9 96.6
1936 74.4 99.9
1937 75.7 100.5
1938 70.2 100.3
1939 73.2 103.1
• Fast Recovery of A, slow recovery of output. Why?
50

Monetary Cycle Models
1

Money, Prices and Output
• What are the eﬀects of changes of money supply on prices and output?
• We will think about two cases:
1. The long run.
2. The short run.
2

Some Facts about Money in the Long Run
Reported by McCandless and Weber (1995) and Rolnick and Weber (1998):
1. There is a high (almost unity) correlation between the rate of growth
of monetary supply and the rate of inflation.
2. There is no correlation between the growth rates of money and real
output.
3. There in no correlation between inflation and real output.
4. Inflation and money growth are higher under fiat money than under
commodity standards.
3

Money in the Long Run
• Economist have a pretty good idea of how to think about this case.
• Take the Fisher Equation (really an accounting identity):
MV = PY
• If V and Y are roughly constant then
gm = gp
• “Inﬂation is always and everywhere a monetary phenomenon”, Fried-
man (1956).
4

Applying the Theory
• Earl J. Hamilton’s (1934) “American Treasure and the Price Revolu-
tion in Spain, 1501-1650”.
• Germany’s Hyperinﬂation.
• Nowadays.
5

Money and the Cycle
• Two observations:
1. Relation between Output and Money growth.
2. Phillips Curve.
• “Conventional Wisdom”:
1. Volcker’s recessions,
2. Friedman and Schwartz (1963), “A Monetary History of the US”.
6

Phillips Curve for the US between 1967−99
Unemployment Rate in Deviation from Natural Rate
InflationRate
−4 −3 −2 −1 0 1 2 3 4 5 6
−0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18

• It is surprisingly diﬃcult to build models of monetary cycle.
• Two models:
1. Lucas imperfect-information model.
2. Sticky prices-wages model.
7

Imperfect-Information Model I
• Household makes labor supply decisions based on the real wage:
w =
W
P
• Let us suppose that the substitution eﬀect dominates in each period
because of intertemporal substitution:
l0(w) > 0
• Household observes W.
8

Imperfect-Information Model II
• Economy is hit by shocks to productivity A (that raise w) and to the
money supply (that raise P).
• With perfect information on P, household just ﬁnds:
w =
W
P
and takes labor supply decisions.
• But what happens if P is not observed (or only with some noise)?
9

Imperfect-Information Model III
• The household observes W going up.
• Signal extraction problem: decide if W goes up because P goes up
(since W = wp) or because w went up.
• Response is diﬀerent. First case labor supply is constant, in the second
it should increase.
10

Imperfect-Information Model IV
• Then money supply surprises aﬀect labor supply and with it total out-
put: a version of the Phillips curve.
• Policy implications:
1. Government may engine an expansion with a surprise.
2. Government cannot get systematic surprises.
3. Time inconsistency problem: Kydland and Prescott (1977).
11

Imperfect-Information Model V
Criticisms:
1. Imperfect-Information assumption. Just get the WSJ.
2. Persistence.
3. Intertemporal Substitution.
4. Wages are acyclical or procyclical.
5. Is temporal inconsistency such a big deal? Sargent (2000)
12

Sticky Prices-Wages Model I
• Prices and Wages are sticky in the short run.
• Diﬀerent reasons:
1. Menu cost, Mankiw (1995).
2. Staggered contracts, Taylor (1979).
• Money can have real eﬀects.
13

Sticky Prices-Wages Model II
• Basic Sticky-prices model.
• A lot of ﬁrms that are monopolistic competitors: mark-ups.
• Firms set prices for several periods in advance:
1. Deterministic: Taylor Pricing.
2. Stochastic: Calvo Pricing.
• Firms are ready to supply any quantity of the good at the given price.
14

Sticky Prices-Wages Model III
• After prices are set up, money shocks occur.
• If the shock is positive, interest rate goes down and demand for goods
goes up.
• Firms produce more, output goes up, unemployment goes down.
• Persistence problem: will ﬁrms change prices soon?
15

Sticky Prices-Wages Model IV
• Case with sticky wages is similar.
• After money shock, prices go up, real wages go down and ﬁrms produce
more.
• Real wages may fail to fall in the long run: eﬃciency wages.
16

Sticky Prices-Wages Model V
• Again we have a Phillips curve.
• Policy implications:
1. Government wants to keep inﬂation low and stable.
2. Government may use monetary policy to stabilize the economy.
• Taylor’s rule (Taylor, 1993, Rothemberg and Woodford, 1997).
• Are monetary rules normative or positive?
17

Sticky Prices-Wages Model V
Criticisms:
1. Why are prices and wages sticky?
2. Persistence problem.
3. Problem interpreting empirical evidence.
4. Wages behavior over the cycle.
5. Size problem: account for 15-25% of the output variability.
18

Coordination Failure Cycles
1

Externalities
• Externalities can be important.
• For instance: how much would you pay for a phone if you have the
ONLY phone on earth?
• Your utility may depend on what other agents do: strategic comple-
mentarities.
• Develop by Diamond (1982) in search models.
2

Coordination
• Coordination issues are also important.
• Driving: UK versus US.
• We have two diﬀerent Equilibria!
3

A Simple Model
• Cooper and John (1988).
• Utility of the agent i
Ui = V (yi, y)
• Best-Response y0
i(y).
• Equilibrium:
y = y0
i(y)
4

Multiple Equilibria
• The function y = y0
i(y) may have several solutions.
• Each solution is a diﬀerent equilibrium.
• Which one will be selected? Opens door to sunspots and self-fulﬁlling
prophecies.
• Equilibria may be pareto-ranked.
5

Example of Multiple Equilibria
• Increasing returns to scale in the production function (Farmer-Guo,
1994).
• If a lot of us work, marginal productivity is high, wages are high and
as a consequence a lot of us want to work: we have an equilibrium.
• If few of us work, marginal productivity is low, wages are low and
as a consequence few of us want to work: we have another, worse,
equilibrium.
• Fluctuations are just jumps from one equilibrium to another.
6

Household Problem I
• Continuum of households i ∈ [0, 1].
• Each household i has a backyard technology:
yit = Atkα
itl1−α
it
• We can rewrite it as:
max E
∞X
t=1
βt−1u (ct, 1 − lt)
ct + kt+1 = yit + (1 − δ) kt, ∀ t > 0
7

Externalities I
• The externality is given by:
At =
µZ
i
kα
itl1−α
it di
¶θ
• Interpretation.
• Alternatives: monopolistic competition.
8

Externalities II
• Since all the agents are identical:
yt = k
α(1+θ)
t l
(1−α)(1+θ)
t = kυ
t l
µ
t
• Increasing returns to scale production function at the aggregate level...
• but constant at the household level.
9

Solving the Model I
• We can solve the model as we did for the Real Business Model.
• Things only get a bit more complicated because we need to allow for
the presence of shocks to beliefs.
• Instead of postulating policy functions of the form (kt+1 − k) = P (kt − k)+
Qzt and (lt − l) = R (kt − k) + Szt, we propose:
(kt+1 − k) = P (kt − k) + Q (ct − c)
(ct+1 − c) = R (kt − k) + S (ct − c) + εt
10

Solving the Model II
• Technical discussion (skip if wanted): we need as many state variables
as stable roots of the system.
• (ct − c) is not really a “pure” state variable.
• We will omit the details in the interest of time saving.
11

Dynamics of the Model I
• Interaction between the labor supply curve and the labor demand
curve.
• Possibility of self-fulﬁlling equilibria.
• We have seen those before: money.
12

Dynamics of the Model II
• Note that we have recurrent shocks to beliefs.
• The dynamics of the model are stochastic.
• This means we need to simulate as in the case of the standard RBC.
13

Deterministic versus Stochastic Dynamics
• This is diﬀerent from models that exhibit Chaos.
• It is relatively easy to build equilibrium models with permanent deter-
ministic cycles (Benhabib-Boldrin, 1992).
• Tent mapping example.
• Why do economist use few models with chaotic dynamics?
14

Evaluation of the Model
• Comparison with standard RBC.
• Policy implications: government may play a role since diﬀerent equi-
libria are pareto-ranked.
• Do we observe big increasing returns to scale in the US economy?
Insuﬃcient variation of the inputs data (Cole and Ohanian (1999)).
15

Introduction to Optimization
1

1 Unconstrained Maximization
• Consider a function f : <N → <.
• We often called this function the objective function.
Deﬁnition 1.1 A set A is an open neighborhood of x if x ∈ A and A is an
open set.
2

Definition 1.2 The vector x∗ ∈ <N is a local maximizer of f (·) if ∃ an
open neighborhood A of x such that for ∀x ∈ A : f (x∗) ≥ f (x)
Definition 1.3 The vector x∗ ∈ <N is a global maximizer of f (·) if for
∀x ∈ <N : f (x∗) ≥ f (x)
• The concepts of local and global minimizer are defined analogously.
• Clearly every global maximizer is a local maximizer but the converse
is not true.
3

Theorem 1.1 (Necessity) Suppose that f (·) is diﬀerentiable and that x∗ ∈
<N is a local maximizer or minimizer of f (·). Then:
∂f (x∗)
∂xn
= 0 ∀n ∈ {1, ..., N}
or using a more concise notation:
∇f (x∗) = 0
4

Proof. Suppose that x∗ is a local maximizer of f (·) but that
∂f(x∗)
∂xn
=
a > 0 (an analog argument holds if a < 0 or for local minimizers). Deﬁne
the vector en ∈ <N as en
n = 1 and en
h = 0 for h 6= n (i.e. having
its nth entry equal to 1 and all the other entries equal to 0). Then,
by the deﬁnition of partial derivative ∃ ε > 0 arbitrarily small such that
[f (x∗ + εen) − f (x∗)] /ε > a/2 > 0. But then f (x∗ + εen) > f (x∗) +
(εa/2) f (x∗), a contradiction with the fact that x∗ is a local maximizer.
5

Deﬁnition 1.4 A vector x∗ ∈ <N is a critical point if ∇f (x∗) = 0
Corollary 1.2 Every local maximizer or minimizer is a critical point.
Corollary 1.3 Some critical points are not local maximizers or minimizers.
Example 1.1 Consider the function f (x1, x2) = x2
1 − x2
2. We have that
∇f(0, 0) = 0 but that point is neither a local maximizer or minimizer.
6

Definition 1.5 The N × N matrix M is negative semidefinite if
z0Mz ≤ 0
for ∀z ∈ <N. If the inequality is strict then M is negative definite.
An analogous definition holds for positive semidefinite and positive definite
matrices.
7

Theorem 1.4 A Matrix M is positive semidefinite (respectively, positive
definite) if and only if the matrix −M is negative semidefinite (respectively,
negative definite).
8

Theorem 1.5 Let M be an N × N matrix.
1. Suppose that M is symmetric. Then M is negative definite if and only if
(−1)r |rMr| > 0 for every r = 1, .., N.
2. Suppose that M is symmetric. Then M is negative semidefinite if and
only if (−1)r |rMπ
r | > 0 for every r = 1, .., N and for every permutation
π = 1, .., N of the indices of rows and columns.
Opposite results will hold for positive semidefinite and positive definite
matrices.
9

Theorem 1.6 (Sufficiency) Suppose that f (·) is twice continuous differen-
tiable and that ∇f (x∗) = 0.
1. If x∗ ∈ <N is a local maximizer then the symmetric N×N matrix D2f (x∗)
is negative semidefinite.
2. If D2f (x∗) is negative definite x∗ is a local maximizer.
Replacing negative for positive the same is true for local minimizers.
10

Proof. (Outline) Take a second order expansion of f (·) around the
local maximizers. Note that the function less the first (constant) term is
negative. Since the linear term is zero by our previous result the second
order term also must be negative. For that to hold D2f (x∗) must be
negative semidefinite (higher order terms can be safely ignored).
Conversely if D2f (x∗) is negative definite all points in a local neighborhood
must have lower values.
11

• Often the condition
∇f (x∗) = 0
is called the ﬁrst order condition and the fact that D2f (x∗) is negative
semideﬁnite a second order condition.
• Finally note that any critical point of a concave function is a global
maximizer for that function and analogously any critical point of a
convex function is a global minimizer.
12

Example 1.2 Consider the function f(x) = −ax2 + bx where a and b are
constant.
The ﬁrst order condition is:
f0(x∗) = −2ax∗ + b = 0 ⇒ x∗ = b/(2a)
and the second order condition is:
f00(x∗) = −2a < 0
Then if a > 0 x∗ = b/(2a) is a global maximum.
13

Example 1.3 Consider the function f(x1, x2) = −ax2
1−bx2
2+cx1x2−x1−
x2 where a, b > 0 and 4ab − c2 > 0.
The ﬁrst order conditions are:
∂f (x∗)
∂x1
= −2ax1 + cx2 − 1 = 0
∂f (x∗)
∂x2
= −2bx2 + cx1 − 1 = 0
with solution:
x∗
1 =
c + 2b
c2 − 4ba
x∗
2 =
c + 2a
c2 − 4ba
14

To check the second order condition note that:
D2f (x∗) =
Ã
−2a c
c −2b
!
clearly negative deﬁnite given our assumptions on the parameters.
15

• If the second order conditions do not hold we need to examine the logic
of the problem to find whether we have a local or global maximizer.
• Also note that since we look at second derivatives for the case where
the objective function is linear we cannot assure sufficiency. Again we
need to study the specific conditions of the problem.
16

2 Constrained Optimization
We study the problem:
max
x∈<N
f(x)
s.t. g1 (x) = b1
...
gM (x) = bM
where f : <N → < and gm : <N → < for m = 1, ..., M.
17

Definition 2.1 The constraint set is
C =
n
x ∈ <N : gm (x) = bm for m = 1, ..., M
o
Definition 2.2 The vector x∗ ∈ C is a local constrained maximizer of f (·)
if ∃ an open neighborhood A of x such that for ∀x ∈ A∩C : f (x∗) ≥ f (x)
Definition 2.3 The vector x∗ ∈ C is a global constrained maximizer of
f (·) if for ∀x ∈ C : f (x∗) ≥ f (x)
The concepts of local and global constrained minimizer are defined analo-
gously.
18

To solve this problem we build an auxiliary function called the Lagrangian:
L (x) = f(x) +
MX
m=1
λm (b − gm (x))
where the numbers λm are called the Lagrange multipliers.
19

Theorem 2.1 Suppose that f : <N → < and gm : <N → < for m =
1, ..., M are diﬀerentiable and that x∗ is a local constrained maximizer.
Then ∃ numbers λm such that:
∇f (x∗) =
MX
m=1
λm∇gm (x)
The theorem shows that the constrained maximizer is a critical point of
the lagrangian function for an appropriate choice of λm.
The Lagrangian multipliers have a nice interpretation as the shadow prices
of the constraints.
20

Example 2.1 Consider the function f(x1, x2) = −ax2
1 − bx2
2 − x1 − x2.
Suppose we want to maximize it subject to the constraint that:
4x1 − x2 = 0
Then we build the lagrangian:
−ax2
1 − bx2
2 − x1 − x2 + λ(x2 − 4x1)
with the ﬁrst order condition:
−2ax1 − 1 = 4λ
−2bx2 − 1 = −λ
and the constraint:
4x1 − x2 = 0
21

i.e. a system of three equations on three unknowns with solution:
x1 =
−5
2 (a + 16b)
x2 =
−20
2 (a + 16b)
λ =
20bx2
2 (a + 16b)
− 1
22

The second order theory for constrained problems is simple: we apply all
previous results regarding the second derivatives matrix to the function
L (x) instead of f(x).
The proof that the second order conditions hold for the previous example
is left to you.
23

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